{"id":345,"date":"2026-07-18T13:03:25","date_gmt":"2026-07-18T13:03:25","guid":{"rendered":"https:\/\/mathpk.com\/?page_id=345"},"modified":"2026-07-18T13:04:26","modified_gmt":"2026-07-18T13:04:26","slug":"mock-test-1","status":"publish","type":"page","link":"https:\/\/mathpk.com\/?page_id=345","title":{"rendered":"Mock Test 1"},"content":{"rendered":"<style>.kb-row-layout-id345_dc5079-c0 > .kt-row-column-wrap{align-content:start;}:where(.kb-row-layout-id345_dc5079-c0 > .kt-row-column-wrap) > .wp-block-kadence-column{justify-content:start;}.kb-row-layout-id345_dc5079-c0 > .kt-row-column-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:var(--global-kb-gap-md, 2rem);padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);grid-template-columns:minmax(0, 1fr);}.kb-row-layout-id345_dc5079-c0 > .kt-row-layout-overlay{opacity:0.30;}@media all and (max-width: 1024px){.kb-row-layout-id345_dc5079-c0 > 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10px;}\n\nfooter.note{text-align:center;color:var(--skip);font-size:12px;margin-top:36px;font-family:var(--mono);}\n\n@media (max-width:640px){\n  .qpanel{padding:20px 18px;}\n  .score-hero{flex-direction:column;text-align:center;}\n  .ch-row{grid-template-columns:1fr;gap:6px;}\n  .ch-pct{text-align:left;}\n}\n<\/style>\n<\/head>\n<body>\n<div class=\"app\">\n\n  <div class=\"hero\">\n    <div class=\"hero-title-block\">\n      <div class=\"eyebrow\">Practice Test \u00b7 [ A ]<\/div>\n      <h1>Matrices &amp; Determinants<\/h1>\n      <p>100 conceptual MCQs for FAST-NU, NUST-NET, UET-ECAT &amp; engineering entry tests.<\/p>\n    <\/div>\n    <div class=\"clock-panel\">\n      <div>\n        <div class=\"clock-face\" id=\"clockFace\">00:00<\/div>\n        <div class=\"clock-label\">elapsed time<\/div>\n      <\/div>\n      <div class=\"clock-btns\">\n        <button id=\"clockStart\">Start<\/button>\n        <button id=\"clockPause\">Pause<\/button>\n        <button id=\"clockReset\">Reset<\/button>\n      <\/div>\n    <\/div>\n  <\/div>\n\n  <div class=\"progress-strip\">\n    <div class=\"progress-text\" id=\"progressText\">0 \/ 100 answered<\/div>\n    <div class=\"progress-track\"><div class=\"progress-fill\" id=\"progressFill\" style=\"width:0%\"><\/div><\/div>\n    <div class=\"legend\">\n      <span><i style=\"background:var(--brand)\"><\/i>Answered<\/span>\n      <span><i style=\"background:var(--amber-soft);border:1px solid var(--amber)\"><\/i>Skipped<\/span>\n      <span><i style=\"background:var(--purple)\"><\/i>Marked<\/span>\n      <span><i style=\"background:var(--paper);border:1px solid var(--rule-strong)\"><\/i>Unseen<\/span>\n    <\/div>\n  <\/div>\n\n  <div class=\"quiz-view\" id=\"quizView\">\n    <div class=\"layout\">\n      <nav class=\"matrix-nav\" id=\"matrixNav\" aria-label=\"Question navigator\">\n        <h2><span class=\"bk\">[ N ]<\/span> Question Map<\/h2>\n        <div id=\"navSections\"><\/div>\n      <\/nav>\n\n      <section class=\"qpanel\" id=\"qpanel\">\n        <div class=\"qhead\">\n          <div>\n            <div class=\"qmeta\" id=\"qSection\"><\/div>\n            <div class=\"qnum\" id=\"qNum\"><\/div>\n          <\/div>\n          <button class=\"mark-btn\" id=\"markBtn\">\u2606 Mark for review<\/button>\n        <\/div>\n        <div class=\"qtext\" id=\"qText\"><\/div>\n        <div class=\"options\" id=\"qOptions\"><\/div>\n        <div class=\"qfooter\">\n          <div class=\"nav-btns\">\n            <button class=\"btn btn-ghost\" id=\"prevBtn\">\u2190 Previous<\/button>\n            <button class=\"btn btn-ghost\" id=\"nextBtn\">Next \u2192<\/button>\n          <\/div>\n          <button class=\"btn btn-submit\" id=\"submitBtn\">Submit Test<\/button>\n        <\/div>\n      <\/section>\n    <\/div>\n  <\/div>\n\n  <div class=\"results\" id=\"resultsView\">\n    <div class=\"score-hero\">\n      <div class=\"gauge-wrap\">\n        <svg width=\"150\" height=\"150\" viewBox=\"0 0 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Confirm submit modal -->\n<div class=\"modal-overlay\" id=\"confirmOverlay\">\n  <div class=\"modal-window\" style=\"max-width:420px;\">\n    <div class=\"modal-header\"><span class=\"tag\">Submit Test<\/span><button class=\"modal-close\" id=\"confirmCloseX\">\u2715<\/button><\/div>\n    <div class=\"modal-body confirm-body\">\n      <p id=\"confirmText\">Are you sure you want to submit?<\/p>\n      <div class=\"confirm-actions\">\n        <button class=\"btn btn-ghost\" id=\"confirmCancel\">Keep Working<\/button>\n        <button class=\"btn btn-submit\" id=\"confirmYes\">Submit Now<\/button>\n      <\/div>\n    <\/div>\n  <\/div>\n<\/div>\n\n<!-- Solution modal -->\n<div class=\"modal-overlay\" id=\"solutionOverlay\">\n  <div class=\"modal-window\">\n    <div class=\"modal-header\"><span class=\"tag\" id=\"solTag\">Question 1 \u00b7 Solution<\/span><button class=\"modal-close\" id=\"solCloseX\">\u2715<\/button><\/div>\n    <div class=\"modal-body\">\n      <div class=\"modal-q\" id=\"solQ\"><\/div>\n      <div id=\"solOpts\"><\/div>\n      <div class=\"modal-tags\" id=\"solTags\"><\/div>\n      <div class=\"explain\" id=\"solExplain\"><\/div>\n    <\/div>\n  <\/div>\n<\/div>\n\n<script id=\"quiz-data\" type=\"application\/json\">[{\"n\": 1, \"section\": \"Matrices & Order\", \"q\": \"A matrix with m rows and n columns is said to have order:\", \"opts\": [\"n \u00d7 m\", \"m \u00d7 n\", \"m + n\", \"m \u2212 n\"], \"correct\": 1, \"exp\": \"Order is always written as rows \u00d7 columns, so m rows and n columns gives order m \u00d7 n.\"}, {\"n\": 2, \"section\": \"Matrices & Order\", \"q\": \"A matrix that has only one row is called a:\", \"opts\": [\"Row matrix (row vector)\", \"Square matrix\", \"Column matrix\", \"Diagonal matrix\"], \"correct\": 0, \"exp\": \"A matrix with exactly one row (order 1 \u00d7 n) is called a row matrix or row vector.\"}, {\"n\": 3, \"section\": \"Matrices & Order\", \"q\": \"Two matrices A and B are equal if and only if:\", \"opts\": [\"They have the same number of rows\", \"They have the same order and every corresponding element is equal\", \"det(A) = det(B)\", \"A + B = 0\"], \"correct\": 1, \"exp\": \"Matrix equality requires identical dimensions and every corresponding entry to match exactly.\"}, {\"n\": 4, \"section\": \"Matrices & Order\", \"q\": \"A matrix in which the number of rows equals the number of columns is called a:\", \"opts\": [\"Rectangular matrix\", \"Row matrix\", \"Identity matrix\", \"Square matrix\"], \"correct\": 3, \"exp\": \"When rows = columns (order n \u00d7 n), the matrix is called square.\"}, {\"n\": 5, \"section\": \"Matrices & Order\", \"q\": \"A matrix of order 2 \u00d7 3 has:\", \"opts\": [\"2 rows and 3 columns\", \"3 rows and 2 columns\", \"6 columns and 2 rows\", \"5 elements in total\"], \"correct\": 0, \"exp\": \"Order 2 \u00d7 3 means 2 rows and 3 columns, giving 6 total elements.\"}, {\"n\": 6, \"section\": \"Types of Matrices\", \"q\": \"A square matrix in which all non-diagonal elements are zero is called a:\", \"opts\": [\"Scalar matrix\", \"Identity matrix\", \"Zero matrix\", \"Diagonal matrix\"], \"correct\": 3, \"exp\": \"Any square matrix with zeros everywhere except the main diagonal is a diagonal matrix.\"}, {\"n\": 7, \"section\": \"Types of Matrices\", \"q\": \"A diagonal matrix in which every diagonal element equals 1 is called a(n):\", \"opts\": [\"Scalar matrix\", \"Symmetric matrix\", \"Identity (unity) matrix\", \"Upper triangular matrix\"], \"correct\": 2, \"exp\": \"A diagonal matrix with all 1's on the diagonal is the identity matrix.\"}, {\"n\": 8, \"section\": \"Types of Matrices\", \"q\": \"In an upper triangular matrix, all elements below the main diagonal are:\", \"opts\": [\"Equal to 1\", \"Non-zero\", \"Zero\", \"Equal to the diagonal elements\"], \"correct\": 2, \"exp\": \"By definition, an upper triangular matrix has zeros in every position below the main diagonal.\"}, {\"n\": 9, \"section\": \"Types of Matrices\", \"q\": \"A scalar matrix is a diagonal matrix in which all diagonal elements are:\", \"opts\": [\"Zero\", \"Different from each other\", \"Equal to 1\", \"Equal to the same constant k\"], \"correct\": 3, \"exp\": \"A scalar matrix is kI \u2014 a diagonal matrix whose diagonal entries all equal the same constant k.\"}, {\"n\": 10, \"section\": \"Types of Matrices\", \"q\": \"The zero matrix (null matrix) O satisfies:\", \"opts\": [\"A + O = A for any matrix A of the same order\", \"A \u00b7 O = A\", \"A + O = O\", \"O is always a square matrix\"], \"correct\": 0, \"exp\": \"O behaves like additive identity: adding the zero matrix leaves any matrix A unchanged.\"}, {\"n\": 11, \"section\": \"Types of Matrices\", \"q\": \"A lower triangular matrix has all elements above the main diagonal equal to:\", \"opts\": [\"1\", \"0\", \"Any real number\", \"The corresponding diagonal element\"], \"correct\": 1, \"exp\": \"A lower triangular matrix is zero everywhere above the main diagonal by definition.\"}, {\"n\": 12, \"section\": \"Types of Matrices\", \"q\": \"Which of the following statements about a rectangular matrix is correct?\", \"opts\": [\"Its number of rows is NOT equal to its number of columns\", \"Its number of rows equals its number of columns\", \"It always has a determinant\", \"It is always a row matrix\"], \"correct\": 0, \"exp\": \"A rectangular matrix simply means rows \u2260 columns, unlike a square matrix.\"}, {\"n\": 13, \"section\": \"Types of Matrices\", \"q\": \"The main diagonal of a matrix A = [a\u1d62\u2c7c] consists of elements where:\", \"opts\": [\"i > j\", \"i < j\", \"i \u2260 j\", \"i = j\"], \"correct\": 3, \"exp\": \"The main diagonal consists of entries where the row index equals the column index (i = j).\"}, {\"n\": 14, \"section\": \"Types of Matrices\", \"q\": \"A matrix with a single column is called a:\", \"opts\": [\"Row vector\", \"Scalar matrix\", \"Column matrix (column vector)\", \"Square matrix\"], \"correct\": 2, \"exp\": \"An n \u00d7 1 matrix, having one column, is called a column matrix or column vector.\"}, {\"n\": 15, \"section\": \"Types of Matrices\", \"q\": \"Which matrix type has exactly the same entries on both sides of the main diagonal, i.e., a\u1d62\u2c7c = a\u2c7c\u1d62?\", \"opts\": [\"Skew-symmetric\", \"Upper triangular\", \"Scalar\", \"Symmetric\"], \"correct\": 3, \"exp\": \"The defining property a\u1d62\u2c7c = a\u2c7c\u1d62 for all i, j is exactly the definition of a symmetric matrix.\"}, {\"n\": 16, \"section\": \"Transpose, Symmetric &#038; Skew-Symmetric\", \"q\": \"The transpose of matrix A, denoted A\u1d40, is obtained by:\", \"opts\": [\"Multiplying all elements by \u22121\", \"Interchanging the rows and columns of A\", \"Replacing each element by its cofactor\", \"Multiplying A by the identity matrix\"], \"correct\": 1, \"exp\": \"Transposing swaps rows and columns: row i of A becomes column i of A\u1d40.\"}, {\"n\": 17, \"section\": \"Transpose, Symmetric &#038; Skew-Symmetric\", \"q\": \"If A is an m \u00d7 n matrix, the order of A\u1d40 is:\", \"opts\": [\"m \u00d7 n\", \"m \u00d7 m\", \"n \u00d7 m\", \"n \u00d7 n\"], \"correct\": 2, \"exp\": \"Transposing an m \u00d7 n matrix flips the dimensions, giving an n \u00d7 m matrix.\"}, {\"n\": 18, \"section\": \"Transpose, Symmetric &#038; Skew-Symmetric\", \"q\": \"A square matrix A is called skew-symmetric if:\", \"opts\": [\"A = A\u1d40\", \"A = \u2212A\u1d40\", \"A + A\u1d40 = I\", \"A \u00b7 A\u1d40 = 0\"], \"correct\": 1, \"exp\": \"Skew-symmetric matrices satisfy A\u1d40 = \u2212A, meaning off-diagonal entries are negatives of their mirror.\"}, {\"n\": 19, \"section\": \"Transpose, Symmetric &#038; Skew-Symmetric\", \"q\": \"If A is a skew-symmetric matrix, then all its diagonal elements must be:\", \"opts\": [\"Equal to 1\", \"Non-zero\", \"Zero\", \"Equal to \u22121\"], \"correct\": 2, \"exp\": \"Since a\u1d62\u1d62 = \u2212a\u1d62\u1d62 must hold for skew-symmetric matrices, every diagonal element must be 0.\"}, {\"n\": 20, \"section\": \"Transpose, Symmetric &#038; Skew-Symmetric\", \"q\": \"For any square matrix A, the matrix A + A\u1d40 is always:\", \"opts\": [\"Skew-symmetric\", \"Singular\", \"The zero matrix\", \"Symmetric\"], \"correct\": 3, \"exp\": \"Taking the transpose of (A + A\u1d40) gives A\u1d40 + A back, so it is always symmetric.\"}, {\"n\": 21, \"section\": \"Matrix Operations\", \"q\": \"Matrix addition is defined only when the two matrices have:\", \"opts\": [\"The same order (same number of rows AND columns)\", \"The same number of rows only\", \"The same determinant value\", \"At least one common element\"], \"correct\": 0, \"exp\": \"Addition is done entrywise, so both matrices must share identical order.\"}, {\"n\": 22, \"section\": \"Matrix Operations\", \"q\": \"For matrices A, B, and C of the same order, which law holds for addition?\", \"opts\": [\"A(B + C) = AB + AC\", \"A + B = A \u00b7 B\", \"A + B = B + A (Commutative)\", \"(AB)C = A(BC)\"], \"correct\": 2, \"exp\": \"Matrix addition is commutative, just like ordinary number addition: A + B = B + A.\"}, {\"n\": 23, \"section\": \"Matrix Operations\", \"q\": \"The product AB of two matrices is defined if and only if:\", \"opts\": [\"A and B are both square matrices\", \"The number of columns of A equals the number of rows of B\", \"A and B have the same order\", \"det(A) = det(B)\"], \"correct\": 1, \"exp\": \"Multiplication requires the inner dimensions to match: columns of A = rows of B.\"}, {\"n\": 24, \"section\": \"Matrix Operations\", \"q\": \"In general, for matrices, which property does NOT hold?\", \"opts\": [\"A(BC) = (AB)C (Associativity)\", \"A(B + C) = AB + AC (Distributivity)\", \"AB = BA (Commutativity)\", \"A \u00b7 I = A\"], \"correct\": 2, \"exp\": \"Matrix multiplication is generally not commutative \u2014 AB usually differs from BA.\"}, {\"n\": 25, \"section\": \"Matrix Operations\", \"q\": \"If A is of order 2 \u00d7 3 and B is of order 3 \u00d7 4, then the order of AB is:\", \"opts\": [\"3 \u00d7 3\", \"2 \u00d7 4\", \"3 \u00d7 4\", \"4 \u00d7 2\"], \"correct\": 1, \"exp\": \"The product takes the outer dimensions: rows of A by columns of B, giving 2 \u00d7 4.\"}, {\"n\": 26, \"section\": \"Matrix Operations\", \"q\": \"Which of the following is true for the transpose of a product of matrices?\", \"opts\": [\"(AB)\u1d40 = B\u1d40A\u1d40\", \"(AB)\u1d40 = A\u1d40B\u1d40\", \"(AB)\u1d40 = AB\u1d40\", \"(AB)\u1d40 = A\u1d40B\"], \"correct\": 0, \"exp\": \"The transpose of a product reverses the order of the factors: (AB)\u1d40 = B\u1d40A\u1d40.\"}, {\"n\": 27, \"section\": \"Matrix Operations\", \"q\": \"For a scalar k and matrix A, which identity is correct?\", \"opts\": [\"(kA)\u1d40 = k\u207b\u00b9A\u1d40\", \"(kA)\u1d40 = kA\u1d40\", \"(kA)\u1d40 = k\u00b2A\u1d40\", \"(kA)\u1d40 = A\u1d40\/k\"], \"correct\": 1, \"exp\": \"Scalar multiplication passes through transposition unchanged: (kA)\u1d40 = kA\u1d40.\"}, {\"n\": 28, \"section\": \"Determinants\", \"q\": \"The determinant is defined only for:\", \"opts\": [\"Square matrices\", \"Row matrices only\", \"Any rectangular matrix\", \"Diagonal matrices only\"], \"correct\": 0, \"exp\": \"A determinant is a scalar value that only exists for square matrices.\"}, {\"n\": 29, \"section\": \"Determinants\", \"q\": \"For A = [[a, b], [c, d]], the determinant det(A) = |A| is:\", \"opts\": [\"ab \u2212 cd\", \"ad + bc\", \"ad \u2212 bc\", \"ac \u2212 bd\"], \"correct\": 2, \"exp\": \"For a 2\u00d72 matrix, the determinant is (product of main diagonal) \u2212 (product of the other diagonal): ad \u2212 bc.\"}, {\"n\": 30, \"section\": \"Determinants\", \"q\": \"If two rows (or two columns) of a matrix are identical, its determinant is:\", \"opts\": [\"Equal to 1\", \"Doubled\", \"Undefined\", \"Zero\"], \"correct\": 3, \"exp\": \"Identical rows\/columns make the rows linearly dependent, forcing the determinant to be zero.\"}, {\"n\": 31, \"section\": \"Determinants\", \"q\": \"If a single row of matrix A is multiplied by scalar k to get matrix B, then det(B) equals:\", \"opts\": [\"k \u00b7 det(A)\", \"det(A) + k\", \"det(A)\/k\", \"det(A)\"], \"correct\": 0, \"exp\": \"Scaling one row by k scales the whole determinant by that same factor k.\"}, {\"n\": 32, \"section\": \"Determinants\", \"q\": \"Which determinant property states that interchanging two rows of a matrix changes the sign of the determinant?\", \"opts\": [\"Scalar multiple property\", \"Row-interchange property\", \"Transpose property\", \"Zero-row property\"], \"correct\": 1, \"exp\": \"This is the row-interchange property: swapping any two rows flips the determinant's sign.\"}, {\"n\": 33, \"section\": \"Determinants\", \"q\": \"For any square matrix A, det(A\u1d40) equals:\", \"opts\": [\"\u2212det(A)\", \"0\", \"det(A)\", \"1\/det(A)\"], \"correct\": 2, \"exp\": \"Transposing a matrix does not change its determinant, so det(A\u1d40) = det(A).\"}, {\"n\": 34, \"section\": \"Determinants\", \"q\": \"The product rule for determinants states that for square matrices A and B:\", \"opts\": [\"det(A + B) = det(A) + det(B)\", \"det(AB) = det(A) \u00b7 det(B)\", \"det(AB) = det(A) + det(B)\", \"det(AB) = det(A) \/ det(B)\"], \"correct\": 1, \"exp\": \"Determinants are multiplicative: det(AB) = det(A) \u00b7 det(B).\"}, {\"n\": 35, \"section\": \"Determinants\", \"q\": \"If a row of matrix A is a scalar multiple of another row of A, then:\", \"opts\": [\"det(A) = 0 (rows are linearly dependent)\", \"A is non-singular\", \"det(A) = 1\", \"det(A) is doubled\"], \"correct\": 0, \"exp\": \"Proportional rows are linearly dependent, which always forces the determinant to zero.\"}, {\"n\": 36, \"section\": \"Cofactors, Singular\/Non-Singular, Adjoint &#038; Inverse\", \"q\": \"The cofactor C\u1d62\u2c7c of element a\u1d62\u2c7c in a square matrix A is defined as:\", \"opts\": [\"C\u1d62\u2c7c = M\u1d62\u2c7c (the minor only)\", \"C\u1d62\u2c7c = (\u22121)\u2071\u207a\u02b2 M\u1d62\u2c7c, where M\u1d62\u2c7c is the minor\", \"C\u1d62\u2c7c = (\u22121)\u2071\u207b\u02b2 M\u1d62\u2c7c\", \"C\u1d62\u2c7c = det(A) \/ M\u1d62\u2c7c\"], \"correct\": 1, \"exp\": \"The cofactor attaches a sign (\u22121)^(i+j) to the minor M\u1d62\u2c7c based on its position.\"}, {\"n\": 37, \"section\": \"Cofactors, Singular\/Non-Singular, Adjoint &#038; Inverse\", \"q\": \"A square matrix A is called singular if:\", \"opts\": [\"A\u1d40 = A\", \"A = I\", \"det(A) \u2260 0\", \"det(A) = 0\"], \"correct\": 3, \"exp\": \"A matrix is singular exactly when its determinant is zero, meaning it has no inverse.\"}, {\"n\": 38, \"section\": \"Cofactors, Singular\/Non-Singular, Adjoint &#038; Inverse\", \"q\": \"A non-singular matrix A satisfies:\", \"opts\": [\"det(A) = 0\", \"A has no inverse\", \"det(A) \u2260 0 and A\u207b\u00b9 exists\", \"A = A\u1d40\"], \"correct\": 2, \"exp\": \"Non-singular means the determinant is nonzero, guaranteeing the inverse A\u207b\u00b9 exists.\"}, {\"n\": 39, \"section\": \"Cofactors, Singular\/Non-Singular, Adjoint &#038; Inverse\", \"q\": \"The adjoint (adjugate) of a square matrix A is defined as:\", \"opts\": [\"The matrix of minors of A\", \"The transpose of the matrix of cofactors of A\", \"The determinant of A times the identity\", \"The inverse of A\"], \"correct\": 1, \"exp\": \"adj(A) is formed by taking the cofactor matrix of A and then transposing it.\"}, {\"n\": 40, \"section\": \"Cofactors, Singular\/Non-Singular, Adjoint &#038; Inverse\", \"q\": \"The formula for the inverse of a non-singular matrix A using the adjoint method is:\", \"opts\": [\"A\u207b\u00b9 = det(A) \u00b7 adj(A)\", \"A\u207b\u00b9 = adj(A) \/ det(A)\", \"A\u207b\u00b9 = adj(A) \u00b7 det(A)\", \"A\u207b\u00b9 = det(A) \/ adj(A)\"], \"correct\": 1, \"exp\": \"The inverse is the adjoint divided by the determinant: A\u207b\u00b9 = adj(A)\/det(A).\"}, {\"n\": 41, \"section\": \"Cofactors, Singular\/Non-Singular, Adjoint &#038; Inverse\", \"q\": \"For a 2 \u00d7 2 matrix A = [[a, b], [c, d]] with det(A) \u2260 0, the inverse A\u207b\u00b9 is:\", \"opts\": [\"(1\/det(A)) [[d, \u2212b], [\u2212c, a]]\", \"(1\/det(A)) [[a, \u2212b], [\u2212c, d]]\", \"(1\/det(A)) [[d, b], [c, a]]\", \"(1\/det(A)) [[\u2212d, b], [c, \u2212a]]\"], \"correct\": 0, \"exp\": \"For a 2\u00d72 matrix, swap the diagonal entries, negate the off-diagonal ones, then divide by det(A).\"}, {\"n\": 42, \"section\": \"Cofactors, Singular\/Non-Singular, Adjoint &#038; Inverse\", \"q\": \"Which of the following is true for the inverse of a product of two non-singular matrices A and B?\", \"opts\": [\"(AB)\u207b\u00b9 = A\u207b\u00b9B\u207b\u00b9\", \"(AB)\u207b\u00b9 = B\u207b\u00b9A\u207b\u00b9\", \"(AB)\u207b\u00b9 = (A\u207b\u00b9 + B\u207b\u00b9)\", \"(AB)\u207b\u00b9 = AB\"], \"correct\": 1, \"exp\": \"Like transposition, inversion reverses order: (AB)\u207b\u00b9 = B\u207b\u00b9A\u207b\u00b9.\"}, {\"n\": 43, \"section\": \"Cofactors, Singular\/Non-Singular, Adjoint &#038; Inverse\", \"q\": \"If A is a non-singular matrix, then det(A\u207b\u00b9) equals:\", \"opts\": [\"det(A)\", \"\u2212det(A)\", \"0\", \"1 \/ det(A)\"], \"correct\": 3, \"exp\": \"Since det(A)\u00b7det(A\u207b\u00b9) = det(I) = 1, it follows that det(A\u207b\u00b9) = 1\/det(A).\"}, {\"n\": 44, \"section\": \"Cofactors, Singular\/Non-Singular, Adjoint &#038; Inverse\", \"q\": \"The identity A \u00b7 A\u207b\u00b9 = A\u207b\u00b9 \u00b7 A equals:\", \"opts\": [\"0 (zero matrix)\", \"A\", \"I (identity matrix)\", \"2A\"], \"correct\": 2, \"exp\": \"By definition of an inverse, multiplying A by A\u207b\u00b9 in either order gives the identity matrix I.\"}, {\"n\": 45, \"section\": \"Cofactors, Singular\/Non-Singular, Adjoint &#038; Inverse\", \"q\": \"For a non-singular matrix A, (A\u1d40)\u207b\u00b9 equals:\", \"opts\": [\"(A\u207b\u00b9)\u1d40\", \"A\u207b\u00b9\", \"A\u1d40\", \"\u2212A\u207b\u00b9\"], \"correct\": 0, \"exp\": \"Transpose and inverse commute: the inverse of the transpose equals the transpose of the inverse.\"}, {\"n\": 46, \"section\": \"Determinant Properties &#038; Row\/Column Ops\", \"q\": \"If every element of one row of matrix A is zero, then det(A) is:\", \"opts\": [\"Indeterminate\", \"Equal to 1\", \"Negative\", \"Zero\"], \"correct\": 3, \"exp\": \"A zero row means every term in the cofactor expansion along it vanishes, so det(A) = 0.\"}, {\"n\": 47, \"section\": \"Determinant Properties &#038; Row\/Column Ops\", \"q\": \"Adding a scalar multiple of one row to another row of a matrix:\", \"opts\": [\"Does NOT change the value of the determinant\", \"Changes the sign of the determinant\", \"Multiplies the determinant by that scalar\", \"Makes the matrix singular\"], \"correct\": 0, \"exp\": \"This elementary row operation leaves the determinant's value completely unchanged.\"}, {\"n\": 48, \"section\": \"Determinant Properties &#038; Row\/Column Ops\", \"q\": \"The elementary row operation that multiplies an entire row by a non-zero constant k:\", \"opts\": [\"Does not change det(A)\", \"Multiplies det(A) by k\", \"Divides det(A) by k\", \"Adds k to det(A)\"], \"correct\": 1, \"exp\": \"Scaling one full row by k scales the determinant by that same factor k.\"}, {\"n\": 49, \"section\": \"Determinant Properties &#038; Row\/Column Ops\", \"q\": \"If A is an n \u00d7 n matrix, then det(kA) equals:\", \"opts\": [\"k \u00b7 det(A)\", \"k\u00b2 \u00b7 det(A)\", \"k\u207f \u00b7 det(A)\", \"det(A)\"], \"correct\": 2, \"exp\": \"Scaling all n rows by k multiplies the determinant by k for each row, giving k\u207f\u00b7det(A).\"}, {\"n\": 50, \"section\": \"Determinant Properties &#038; Row\/Column Ops\", \"q\": \"The cofactor expansion (Laplace expansion) of det(A) along any row i gives:\", \"opts\": [\"A different value for each row chosen\", \"The same value regardless of which row or column is chosen\", \"The value only for row 1\", \"Zero for non-square matrices\"], \"correct\": 1, \"exp\": \"Laplace expansion is consistent \u2014 expanding along any row or column yields the same determinant.\"}, {\"n\": 51, \"section\": \"Determinant Properties &#038; Row\/Column Ops\", \"q\": \"If A and B are square matrices of the same order, which is always true?\", \"opts\": [\"det(AB) = det(A) \u00b7 det(B)\", \"det(A \u2212 B) = det(A) \u2212 det(B)\", \"det(A + B) = det(A) + det(B)\", \"det(A\u1d40) = \u2212det(A)\"], \"correct\": 0, \"exp\": \"Only the multiplicative property det(AB) = det(A)\u00b7det(B) always holds in general.\"}, {\"n\": 52, \"section\": \"Determinant Properties &#038; Row\/Column Ops\", \"q\": \"Swapping two columns of a matrix A to get matrix B results in:\", \"opts\": [\"det(B) = 2 \u00b7 det(A)\", \"det(B) = det(A)\", \"det(B) = 0\", \"det(B) = \u2212det(A)\"], \"correct\": 3, \"exp\": \"Just like row swaps, interchanging two columns flips the sign of the determinant.\"}, {\"n\": 53, \"section\": \"Echelon Form, RREF &#038; Rank\", \"q\": \"A matrix is in row echelon form (REF) when:\", \"opts\": [\"All elements are zero\", \"All non-zero rows are above any rows of all zeros, and the leading entry of each non-zero row is to the right of the leading entry of the row above it\", \"All diagonal elements are 1\", \"The matrix is square and upper triangular with 1s on the diagonal\"], \"correct\": 1, \"exp\": \"REF requires a staircase pattern: zero rows at the bottom, and each pivot further right than the one above.\"}, {\"n\": 54, \"section\": \"Echelon Form, RREF &#038; Rank\", \"q\": \"In reduced row echelon form (RREF), the leading entry (pivot) of each non-zero row must be:\", \"opts\": [\"1, and it must be the only non-zero entry in its column\", \"Equal to the row number\", \"Any non-zero value\", \"Greater than all elements in that row\"], \"correct\": 0, \"exp\": \"RREF requires each pivot to equal 1 and to be the sole nonzero entry in its column.\"}, {\"n\": 55, \"section\": \"Echelon Form, RREF &#038; Rank\", \"q\": \"The rank of a matrix A is defined as:\", \"opts\": [\"The number of rows of A\", \"The number of non-zero rows in the row echelon form of A\", \"det(A)\", \"The number of columns of A\"], \"correct\": 1, \"exp\": \"Rank equals the count of nonzero rows once the matrix is reduced to echelon form.\"}, {\"n\": 56, \"section\": \"Echelon Form, RREF &#038; Rank\", \"q\": \"The rank of an n \u00d7 n identity matrix I\u2099 is:\", \"opts\": [\"0\", \"n \u2212 1\", \"n\", \"n\u00b2\"], \"correct\": 2, \"exp\": \"Every row of the identity matrix is nonzero and independent, so its rank equals n.\"}, {\"n\": 57, \"section\": \"Echelon Form, RREF &#038; Rank\", \"q\": \"If the rank of a 3 \u00d7 3 matrix A equals 3, then A is:\", \"opts\": [\"Singular (det(A) = 0)\", \"Non-singular (det(A) \u2260 0) and A\u207b\u00b9 exists\", \"A zero matrix\", \"A scalar matrix with zero diagonal\"], \"correct\": 1, \"exp\": \"Full rank for a square matrix means its rows are independent, so det(A) \u2260 0 and A\u207b\u00b9 exists.\"}, {\"n\": 58, \"section\": \"Echelon Form, RREF &#038; Rank\", \"q\": \"The rank of the zero matrix (all elements zero) is:\", \"opts\": [\"1\", \"Equal to the number of rows\", \"Undefined\", \"0\"], \"correct\": 3, \"exp\": \"A zero matrix has no nonzero rows in echelon form, so its rank is 0.\"}, {\"n\": 59, \"section\": \"Echelon Form, RREF &#038; Rank\", \"q\": \"For an m \u00d7 n matrix A, rank(A) cannot exceed:\", \"opts\": [\"m \u00b7 n\", \"m + n\", \"min(m, n)\", \"max(m, n)\"], \"correct\": 2, \"exp\": \"Rank is bounded by the smaller of the row count and column count: min(m, n).\"}, {\"n\": 60, \"section\": \"Echelon Form, RREF &#038; Rank\", \"q\": \"Performing elementary row operations on a matrix:\", \"opts\": [\"Preserves (does not change) the rank\", \"Changes the rank only if a row is deleted\", \"Always changes the rank\", \"Doubles the rank\"], \"correct\": 0, \"exp\": \"Elementary row operations are reversible and preserve the row space, so rank stays the same.\"}, {\"n\": 61, \"section\": \"Cramer's Rule\", \"q\": \"Cramer's rule can be applied to solve a system AX = B only when:\", \"opts\": [\"A is square and det(A) \u2260 0\", \"det(A) = 0\", \"A is a rectangular matrix\", \"B is the zero vector\"], \"correct\": 0, \"exp\": \"Cramer's rule needs a square coefficient matrix with a nonzero determinant to divide by.\"}, {\"n\": 62, \"section\": \"Cramer's Rule\", \"q\": \"In Cramer's rule for the system AX = B, the value of variable x\u2c7c is:\", \"opts\": [\"x\u2c7c = det(A) \/ det(A\u2c7c)\", \"x\u2c7c = det(A\u2c7c) \/ det(A), where A\u2c7c is A with column j replaced by B\", \"x\u2c7c = det(A\u2c7c) \u00b7 det(A)\", \"x\u2c7c = det(B) \/ det(A)\"], \"correct\": 1, \"exp\": \"Each unknown is the ratio det(A\u2c7c)\/det(A), with column j of A swapped for B.\"}, {\"n\": 63, \"section\": \"Cramer's Rule\", \"q\": \"For a 2 \u00d7 2 system ax + by = e, cx + dy = f with det(A) \u2260 0, Cramer's rule gives x as:\", \"opts\": [\"x = (ed \u2212 bf) \/ (ad \u2212 bc)\", \"x = (af \u2212 ce) \/ (ad \u2212 bc)\", \"x = (ad \u2212 bc) \/ (ed \u2212 bf)\", \"x = (bf \u2212 ed) \/ (ad \u2212 bc)\"], \"correct\": 0, \"exp\": \"Replacing the x-column with (e, f) and dividing by det(A) gives x = (ed \u2212 bf)\/(ad \u2212 bc).\"}, {\"n\": 64, \"section\": \"Cramer's Rule\", \"q\": \"If det(A) = 0 in a system AX = B, Cramer's rule:\", \"opts\": [\"Still gives a unique solution\", \"Gives infinitely many solutions always\", \"Cannot be applied; the system may be inconsistent or have infinitely many solutions\", \"Gives the zero solution X = 0\"], \"correct\": 2, \"exp\": \"With det(A) = 0 division is undefined, so Cramer's rule breaks down and other methods are needed.\"}, {\"n\": 65, \"section\": \"Cramer's Rule\", \"q\": \"Cramer's rule is most practically useful when:\", \"opts\": [\"The system has more equations than unknowns\", \"The system is homogeneous\", \"The coefficient matrix is singular\", \"The system is small (2 \u00d7 2 or 3 \u00d7 3) with a non-zero determinant\"], \"correct\": 3, \"exp\": \"Cramer's rule is efficient mainly for small systems where determinants are quick to compute.\"}, {\"n\": 66, \"section\": \"Systems of Linear Equations\", \"q\": \"A system of linear equations AX = B is called homogeneous if:\", \"opts\": [\"All coefficients are equal\", \"B = 0 (the right-hand side is the zero vector)\", \"det(A) = 0\", \"The system has a unique solution\"], \"correct\": 1, \"exp\": \"A homogeneous system has zero on the right-hand side of every equation: AX = 0.\"}, {\"n\": 67, \"section\": \"Systems of Linear Equations\", \"q\": \"A homogeneous system AX = 0 always has:\", \"opts\": [\"At least the trivial solution X = 0\", \"Exactly one non-trivial solution\", \"No solution\", \"Infinitely many solutions only\"], \"correct\": 0, \"exp\": \"X = 0 always satisfies AX = 0, so a homogeneous system is always consistent with the trivial solution.\"}, {\"n\": 68, \"section\": \"Systems of Linear Equations\", \"q\": \"A system AX = B is called consistent if:\", \"opts\": [\"det(A) = 0\", \"It has at least one solution\", \"It has no solution\", \"A is a non-square matrix\"], \"correct\": 1, \"exp\": \"Consistency simply means the system has at least one solution.\"}, {\"n\": 69, \"section\": \"Systems of Linear Equations\", \"q\": \"A system AX = B is inconsistent if:\", \"opts\": [\"It has exactly one solution\", \"It has infinitely many solutions\", \"rank(A) \u2260 rank([A|B]) (augmented matrix)\", \"A is non-singular\"], \"correct\": 2, \"exp\": \"A mismatch between rank(A) and rank of the augmented matrix means no solution exists.\"}, {\"n\": 70, \"section\": \"Systems of Linear Equations\", \"q\": \"For a non-homogeneous system AX = B with det(A) \u2260 0, the system has:\", \"opts\": [\"Exactly one unique solution\", \"No solution\", \"Infinitely many solutions\", \"Only the trivial solution\"], \"correct\": 0, \"exp\": \"A nonzero determinant guarantees A\u207b\u00b9 exists, giving exactly one solution X = A\u207b\u00b9B.\"}, {\"n\": 71, \"section\": \"Systems of Linear Equations\", \"q\": \"For a homogeneous system AX = 0, a non-trivial solution (X \u2260 0) exists only if:\", \"opts\": [\"det(A) \u2260 0\", \"A is the identity matrix\", \"det(A) = 0\", \"rank(A) = n (number of unknowns)\"], \"correct\": 2, \"exp\": \"Non-trivial solutions require the columns to be dependent, which happens only when det(A) = 0.\"}, {\"n\": 72, \"section\": \"Systems of Linear Equations\", \"q\": \"The augmented matrix [A | B] of the system AX = B is formed by:\", \"opts\": [\"Multiplying A and B\", \"Appending column vector B to the right of coefficient matrix A\", \"Replacing columns of A with B\", \"Adding matrix B to matrix A\"], \"correct\": 1, \"exp\": \"The augmented matrix simply attaches B as an extra column beside coefficient matrix A.\"}, {\"n\": 73, \"section\": \"Systems of Linear Equations\", \"q\": \"For the system AX = B, if rank(A) = rank([A|B]) = n (number of unknowns), the system has:\", \"opts\": [\"No solution\", \"A unique solution\", \"Infinitely many solutions\", \"Only the trivial solution\"], \"correct\": 1, \"exp\": \"Equal ranks matching the number of unknowns means the system has exactly one solution.\"}, {\"n\": 74, \"section\": \"Systems of Linear Equations\", \"q\": \"For the system AX = B, if rank(A) = rank([A|B]) < n, the system has:\", \"opts\": [\"No solution\", \"Exactly two solutions\", \"A unique solution\", \"Infinitely many solutions\"], \"correct\": 3, \"exp\": \"Matching ranks below the number of unknowns leaves free variables, giving infinitely many solutions.\"}, {\"n\": 75, \"section\": \"Systems of Linear Equations\", \"q\": \"Which method converts an augmented matrix [A|B] directly to its reduced row echelon form to read off the solution?\", \"opts\": [\"Gauss-Jordan elimination\", \"Adjoint (inverse) method\", \"Cramer's rule\", \"LU decomposition\"], \"correct\": 0, \"exp\": \"Gauss-Jordan elimination reduces [A|B] all the way to RREF so solutions can be read off directly.\"}, {\"n\": 76, \"section\": \"Matrix Inversion &#038; Gauss-Jordan\", \"q\": \"The matrix inversion method solves AX = B using:\", \"opts\": [\"X = A \u00b7 B\", \"X = B \u00b7 A\u207b\u00b9\", \"X = A\u207b\u00b9 \u00b7 B, provided det(A) \u2260 0\", \"X = A \/ B\"], \"correct\": 2, \"exp\": \"Multiplying both sides of AX = B by A\u207b\u00b9 on the left gives X = A\u207b\u00b9B.\"}, {\"n\": 77, \"section\": \"Matrix Inversion &#038; Gauss-Jordan\", \"q\": \"The matrix inversion method requires that the coefficient matrix A be:\", \"opts\": [\"Symmetric\", \"Non-singular (det(A) \u2260 0)\", \"Diagonal\", \"Rectangular\"], \"correct\": 1, \"exp\": \"The inverse A\u207b\u00b9 only exists when A is non-singular, so this is a strict requirement.\"}, {\"n\": 78, \"section\": \"Matrix Inversion &#038; Gauss-Jordan\", \"q\": \"The Gauss-Jordan method finds A\u207b\u00b9 by applying row operations to the augmented matrix:\", \"opts\": [\"[A | 0]\", \"[A | A]\", \"[A | I], reducing it to [I | A\u207b\u00b9]\", \"[I | A]\"], \"correct\": 2, \"exp\": \"Starting from [A | I] and reducing A to I transforms the identity side into A\u207b\u00b9.\"}, {\"n\": 79, \"section\": \"Matrix Inversion &#038; Gauss-Jordan\", \"q\": \"In the Gauss-Jordan method for solving AX = B, we perform row operations on [A|B] until A becomes:\", \"opts\": [\"An identity matrix I (reduced row echelon form)\", \"A lower triangular matrix\", \"An upper triangular matrix\", \"A diagonal matrix\"], \"correct\": 0, \"exp\": \"Gauss-Jordan reduces the coefficient side fully to the identity matrix, revealing X directly.\"}, {\"n\": 80, \"section\": \"Matrix Inversion &#038; Gauss-Jordan\", \"q\": \"Gaussian elimination reduces a matrix to:\", \"opts\": [\"Reduced row echelon form (RREF)\", \"Row echelon form (REF) with back-substitution\", \"Diagonal form\", \"Lower triangular form\"], \"correct\": 1, \"exp\": \"Gaussian elimination stops at echelon form, then back-substitution finishes solving the system.\"}, {\"n\": 81, \"section\": \"Matrix Inversion &#038; Gauss-Jordan\", \"q\": \"If during Gauss-Jordan elimination a row of the form [0 0 \u2026 0 | c] (c \u2260 0) appears, the system is:\", \"opts\": [\"Consistent with a unique solution\", \"Consistent with infinitely many solutions\", \"Inconsistent (no solution)\", \"Homogeneous\"], \"correct\": 2, \"exp\": \"A row of all zeros equaling a nonzero constant is a contradiction, so the system has no solution.\"}, {\"n\": 82, \"section\": \"Matrix Inversion &#038; Gauss-Jordan\", \"q\": \"The Gauss-Jordan method is an extension of Gaussian elimination that eliminates unknowns:\", \"opts\": [\"Only below each pivot\", \"Above and below each pivot, reducing to RREF\", \"Only from the last row upward\", \"Using cofactor expansion\"], \"correct\": 1, \"exp\": \"Gauss-Jordan clears entries both above and below each pivot, going all the way to RREF.\"}, {\"n\": 83, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"Matrices are used in computer graphics primarily to perform:\", \"opts\": [\"Data sorting\", \"Transformations such as rotation, scaling, and translation of images\", \"Encryption of audio files only\", \"Solving differential equations only\"], \"correct\": 1, \"exp\": \"Graphics engines use matrices to rotate, scale, and translate points and objects efficiently.\"}, {\"n\": 84, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"If A is any square matrix, then A + (\u2212A) equals:\", \"opts\": [\"I\", \"2A\", \"A\", \"O (zero matrix)\"], \"correct\": 3, \"exp\": \"Adding a matrix to its negative cancels every entry, producing the zero matrix.\"}, {\"n\": 85, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"For a 3 \u00d7 3 matrix A, if one row is a linear combination of the other two rows, then:\", \"opts\": [\"det(A) = 0\", \"A is non-singular\", \"rank(A) = 3\", \"det(A) = 3\"], \"correct\": 0, \"exp\": \"Linear dependence among rows means the rows aren't independent, forcing det(A) = 0.\"}, {\"n\": 86, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"The number of elements in a matrix of order m \u00d7 n is:\", \"opts\": [\"m + n\", \"m \u2212 n\", \"m\/n\", \"m \u00b7 n\"], \"correct\": 3, \"exp\": \"Total entries equal rows times columns: m multiplied by n.\"}, {\"n\": 87, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"If A\u00b2 = A for a square matrix A, then A is called a:\", \"opts\": [\"Involutory matrix\", \"Idempotent matrix\", \"Nilpotent matrix\", \"Orthogonal matrix\"], \"correct\": 1, \"exp\": \"A matrix satisfying A\u00b2 = A is called idempotent, since repeated multiplication leaves it unchanged.\"}, {\"n\": 88, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"det(I) for any n \u00d7 n identity matrix I\u2099 equals:\", \"opts\": [\"n\", \"0\", \"1\", \"n!\"], \"correct\": 2, \"exp\": \"The identity matrix always has determinant 1, regardless of its size n.\"}, {\"n\": 89, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"If A is a 3 \u00d7 3 matrix with det(A) = 5, then det(2A) equals:\", \"opts\": [\"10\", \"40\", \"25\", \"15\"], \"correct\": 1, \"exp\": \"For an n\u00d7n matrix, det(kA) = k\u207f\u00b7det(A); here n=3, so det(2A) = 2\u00b3\u00b75 = 40.\"}, {\"n\": 90, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"The minor M\u1d62\u2c7c of element a\u1d62\u2c7c in a square matrix A is:\", \"opts\": [\"The cofactor of a\u1d62\u2c7c\", \"The determinant of the sub-matrix obtained by deleting row i and column j\", \"The element a\u1d62\u2c7c itself\", \"The adjoint of A\"], \"correct\": 1, \"exp\": \"A minor is the determinant left after deleting the row and column containing that element.\"}, {\"n\": 91, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"Which of the following matrices does NOT have an inverse?\", \"opts\": [\"[[1, 2], [3, 4]]\", \"[[2, 4], [1, 3]]\", \"[[2, 4], [1, 2]]\", \"[[3, 1], [2, 1]]\"], \"correct\": 2, \"exp\": \"For [[2,4],[1,2]], det = 2\u00b72 \u2212 4\u00b71 = 0, so it's singular and has no inverse.\"}, {\"n\": 92, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"The system x + y = 3, 2x + 2y = 7 is:\", \"opts\": [\"Consistent with unique solution\", \"Homogeneous\", \"Consistent with infinitely many solutions\", \"Inconsistent (no solution)\"], \"correct\": 3, \"exp\": \"Scaling the first equation by 2 gives 2x+2y=6, which contradicts 2x+2y=7, so no solution exists.\"}, {\"n\": 93, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"If A is an n \u00d7 n non-singular matrix, the system AX = 0 has:\", \"opts\": [\"No solution\", \"Infinitely many solutions\", \"Only the trivial solution X = 0\", \"Exactly n solutions\"], \"correct\": 2, \"exp\": \"A non-singular coefficient matrix means the only solution to AX = 0 is the trivial one.\"}, {\"n\": 94, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"For matrices A and B of the same order, (A + B)\u1d40 equals:\", \"opts\": [\"A\u1d40 + B\u1d40\", \"A\u1d40 \u2212 B\u1d40\", \"B\u1d40 + A\u1d40 only if A = B\", \"(A\u1d40)(B\u1d40)\"], \"correct\": 0, \"exp\": \"Transposition distributes over addition: (A + B)\u1d40 = A\u1d40 + B\u1d40.\"}, {\"n\": 95, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"Which statement correctly describes the echelon form of a matrix?\", \"opts\": [\"All elements are 1 or 0\", \"Pivots are all equal to 1\", \"It is a staircase-shaped pattern where each row's leading entry is to the right of the row above\", \"All rows are equal\"], \"correct\": 2, \"exp\": \"Echelon form has a staircase shape, with each pivot positioned further right than the one above it.\"}, {\"n\": 96, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"In solving AX = B by the matrix inversion method, if A is singular, then:\", \"opts\": [\"X = A \u00b7 B still works\", \"X = 0 always\", \"We must use Cramer's rule instead\", \"This method cannot be applied because A\u207b\u00b9 does not exist\"], \"correct\": 3, \"exp\": \"A singular matrix has no inverse, so the inversion method simply cannot be used.\"}, {\"n\": 97, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"Any matrix A can be written as the sum of a symmetric and a skew-symmetric matrix as:\", \"opts\": [\"A = (A + A\u1d40)\/2 + (A \u2212 A\u1d40)\/2\", \"A = A\u1d40 + (A \u2212 A\u1d40)\", \"A = (A \u2212 A\u1d40)\/2 + (A + A\u1d40)\/2\", \"Both A and C are correct\"], \"correct\": 3, \"exp\": \"Both orderings are algebraically identical, since addition is commutative \u2014 so both A and C hold.\"}, {\"n\": 98, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"If A and B are both n \u00d7 n non-singular matrices, then (AB)\u207b\u00b9 equals:\", \"opts\": [\"A\u207b\u00b9B\u207b\u00b9\", \"B\u207b\u00b9A\u207b\u00b9\", \"(A\u207b\u00b9 + B\u207b\u00b9)\", \"AB\u1d40\"], \"correct\": 1, \"exp\": \"As with transposition, inversion reverses multiplication order: (AB)\u207b\u00b9 = B\u207b\u00b9A\u207b\u00b9.\"}, {\"n\": 99, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"Matrices find application in cryptography because:\", \"opts\": [\"They can sort alphabetical data\", \"An encoding matrix E and its inverse E\u207b\u00b9 can encrypt and decrypt messages\", \"They always produce prime numbers\", \"They reduce the size of data\"], \"correct\": 1, \"exp\": \"An invertible encoding matrix scrambles a message, and its inverse recovers the original text.\"}, {\"n\": 100, \"section\": \"Applications &#038; Mixed Concepts\", \"q\": \"The rank of a matrix equals the number of unknowns in AX = B. 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