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GRE Matrices and Determinants — practice questions — page 2

Explore 36 free Matrices and Determinants questions in GRE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Matrices and Determinants questions and exercises with answers

Showing questions 31–36 of 36.

Matrix multiplication is:Read the answer and worked solution →An ORTHOGONAL matrix A satisfies:Read the answer and worked solution →The determinant of the 2×2 matrix [[a, b], [c, d]] equals:Read the answer and worked solution →For matrices A (3×2) and B (2×4), the product AB has order:Read the answer and worked solution →For the system Ax = b, if rank(A) < rank(augmented matrix A|b), the system has:Read the answer and worked solution →The transpose of the product (AB)^T equals:Read the answer and worked solution →

Worked practice questions

Matrix multiplication is:

  1. Associative + distributive, NOT commutative in general
  2. Always commutative for any square matrix size
  3. Never associative under any circumstance
  4. Commutative for diagonal matrices only, anywhere
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Correct answer: A. Associative + distributive, NOT commutative in general

Matrix multiplication is associative (AB)C = A(BC) and distributive A(B+C) = AB + AC. **But it is NOT generally commutative:** AB ≠ BA. Diagonal matrices commute with each other identity commutes with everything.

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An ORTHOGONAL matrix A satisfies:

  1. $A + A^T = 0$ (skew-symmetric)
  2. $A^T A = I$ (transpose equals inverse)
  3. $\det(A) = 0$ (singular always)
  4. $A^2 = I$ (involution always)
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Correct answer: B. $A^T A = I$ (transpose equals inverse)

Orthogonal: A^T A = I, so A^(-1) = A^T. Determinant = ±1. Used for rotations and reflections (preserve length + angle).

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The determinant of the 2×2 matrix [[a, b], [c, d]] equals:

  1. $ad + bc$ (sum of products)
  2. $a + d - b - c$ (sum and difference)
  3. $abcd$ (product of all)
  4. $ad - bc$ (cross-multiplication difference)
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Correct answer: D. $ad - bc$ (cross-multiplication difference)

2×2 determinant: ad − bc. This is the foundation for larger matrices, use cofactor expansion (Laplace's rule) or row reduction.

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More Mathematics topics

MathematicsStatistics and ProbabilityArithmetic ProgressionSets and RelationsQuadratic EquationsReal Numbers and PolynomialsUnit 1A1: Relations, Functions and GraphsUnit 6B: Definite Integration & Riemann Sums