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Quizzes > Mathematics

Test Your Matrix Mastery with Our Free Matrices Quiz

Ready for a Matrix Multiplication Quiz? Dive into this Matrices Practice Test!

Editorial: Review CompletedCreated By: Kristen LopezUpdated Aug 26, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art geometric grid and numbers representing matrix multiplication determinants transformations on sky blue background.

This matrices quiz helps you practice matrix multiplication, determinants, and transformations so you can spot gaps before a test. Work through clear problems, then see how matrices show up in real-world uses and try one tougher challenge question to stretch your skills.

Given A is 2x3 and B is 3x4, what is the size of AB?
5x7
2x4 - Explanation: Inner dimensions (3) match; result takes the outer dimensions 2 and 4.
3x3
4x2
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Which condition is required for matrix addition A + B to be defined?
A must be square
det(A)=det(B)
A and B must have the same dimensions - Explanation: Addition is entrywise and requires identical shapes.
A and B must be invertible
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What is the identity matrix I_n characterized by?
It satisfies AI_n = I_nA = A for any conformable A - Explanation: Identity acts as a multiplicative identity.
All entries are ones
It has zero determinant for all n
Its transpose is different from itself
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Which statement about matrix multiplication is correct?
It is generally not commutative: AB != BA - Explanation: Order matters except for special cases.
It is always commutative: AB = BA
It is never associative
It distributes only on the right, not on the left
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For a square matrix A, which is true about det(I_n)?
det(I_n) = 1 - Explanation: Identity has ones on the diagonal; product of diagonal entries is 1.
det(I_n) = -1
det(I_n) = n
det(I_n) = 0
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Which operation does not change the determinant of a matrix?
Adding a multiple of one row to another - Explanation: Row replacement leaves det unchanged.
Swapping two rows
Multiplying a row by 3
Transposing then negating entries
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If A is invertible, which is true about (A^T)^{-1}?
(A^T)^{-1} = (A^{-1})^T - Explanation: Taking inverses and transposes commute.
(A^T)^{-1} = A
(A^T)^{-1} = A^2
(A^T)^{-1} = -A^{-1}
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Which matrices are orthogonal?
Matrices Q with Q^T Q = I - Explanation: Columns form an orthonormal set.
Matrices with determinant 0
Any symmetric matrix
Matrices Q with QQ = I
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For square A and B of same size, which is generally true?
tr(AB) = tr(A) + tr(B)
AB = BA
tr(AB) = tr(BA) - Explanation: Trace is invariant under cyclic permutations.
det(AB) = tr(A) tr(B)
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What is the rank of a matrix?
The number of nonzero entries
The dimension of its column space (or row space) - Explanation: Rank equals number of linearly independent columns.
The sum of diagonal entries
The product of singular values
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Cayley-Hamilton theorem asserts what about a square matrix A?
det(A) = 1
A is similar to a diagonal matrix
A satisfies its own characteristic polynomial p_A(A) = 0 - Explanation: Substituting A into p_A yields the zero matrix.
A has real eigenvalues
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Which statement about similar matrices A and B is true?
They must be diagonal
They have the same characteristic polynomial - Explanation: Similarity preserves eigenvalues and multiplicities.
They have identical eigenvectors
They are equal entrywise
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For positive definite symmetric matrix A, which holds for all nonzero x?
Ax = 0
x^T A x > 0 - Explanation: Definition of positive definiteness.
det(A) < 0
x^T A x = 0
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What does Cholesky factorization produce for SPD A?
A = LU with U lower triangular
A = LL^T with L lower triangular and positive diagonal - Explanation: Cholesky exists and is unique for SPD matrices.
A = QR with Q lower triangular
A = UΣV^T with U, V diagonal
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Which is true about the determinant under scalar multiplication cA for n x n A?
det(cA) = c^n det(A) - Explanation: Each row scaled by c multiplies det by c.
det(cA) = c det(A)
det(cA) = det(A)/c
det(cA) = c^{n-1} det(A)
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What is the Kronecker product A ⊗ B of matrices A (m x n) and B (p x q)?
An (mp) x (nq) block matrix where each entry a_ij is replaced by a_ij B - Explanation: Definition of Kronecker product.
An m x n matrix of products of corresponding entries
The matrix AB of size m x q
A direct sum of A and B
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Which is true about the spectral theorem in R^n?
Every skew-symmetric matrix is diagonalizable over R
Every real matrix is orthogonally diagonalizable
Only positive definite matrices are diagonalizable
A real symmetric matrix is orthogonally diagonalizable - Explanation: There exists Q orthogonal with Q^T A Q diagonal.
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If A is 2x2 rotation by theta and B is 2x2 rotation by phi, what is AB?
A rotation by theta + phi - Explanation: Rotations in the plane add angles under composition.
A rotation by theta - phi
A shear
A reflection
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What are the singular values of A?
The roots of det(A - tI)
The diagonal entries of A
The eigenvalues of A
The square roots of eigenvalues of A^T A - Explanation: By definition of SVD singular values.
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For an orthogonal projection P onto column space of A (full column rank), which formula is correct?
P = A^T A
P = A(A^T A)^{-1} A^T - Explanation: Normal-equations projector.
P = A^T(A A^T)^{-1} A
P = (A^T A)^{-1}
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Study Outcomes

  1. Understand Matrix Multiplication -

    Learn to carry out matrix multiplication step by step and interpret the results within the matrix multiplication quiz context.

  2. Compute Determinants Accurately -

    Master the calculation of determinants for various matrix sizes to assess properties like singularity and volume scaling.

  3. Analyze Geometric Transformations -

    Apply matrices to represent and analyze linear transformations such as rotations, reflections, and scalings in two-dimensional space.

  4. Assess Matrix Invertibility -

    Evaluate when a matrix is invertible and practice finding inverse matrices to solve inverse-related problems in this matrices practice test.

  5. Solve Linear Systems Efficiently -

    Use row operations and inverse matrices to solve systems of linear equations, reinforcing techniques from the matrix operations quiz.

Cheat Sheet

  1. Dimension Compatibility for Multiplication -

    Matrix multiplication requires the number of columns in the first matrix to match the number of rows in the second, e.g., a 2×3 matrix times a 3×4 matrix yields a 2×4 result. This fundamental rule is a staple of any matrix multiplication quiz, so remember the mnemonic "rows to columns, then the product unfolds" to avoid mismatch errors.

  2. Determinant Calculation and Interpretation -

    The determinant of a 2×2 matrix [a b; c d] is ad - bc, while for a 3×3 you can apply Sarrus' Rule by summing diagonals and subtracting counter-diagonals. Determinants measure scaling factors and orientation flips in transformations, which you'll often encounter in a matrices practice test. Keep the formula ad - bc at your fingertips for quick checks in any matrix operations quiz.

  3. Inverse Matrices and Cramer's Rule -

    If det(A)≠0 for a 2×2 matrix A, its inverse is (1/det(A))×[d - b; - c a], a handy formula to master for solving systems in a linear algebra self study quiz. In higher dimensions, you can use the adjugate method or row-reduction to find A❻¹, ensuring you cross-verify by checking AA❻¹=I. Applying Cramer's Rule ties determinants directly to solution variables, reinforcing the role of invertibility in system-solving.

  4. Eigenvalues and Eigenvectors in Transformations -

    An eigenvalue λ and eigenvector v satisfy Av=λv, indicating how the linear transformation scales v along its direction. Recognizing this in a matrices quiz helps you see how rotations or stretches act on spaces, especially when diagonalizing A simplifies computations. Recall the phrase "treasure vectors get treasure scalars" to link "eigen" (German for "own") with their "own" scaling factors.

  5. Rank, Row Reduction, and System Solutions -

    The rank of a matrix equals its number of pivot positions after row-reduction, dictating the dimension of its column space and solution existence for Ax=b. A full-rank square matrix guarantees a unique solution, a concept frequently tested in a matrix operations quiz to connect linear independence and solvability. Use the Rank-Nullity Theorem (rank+nullity=n) as a quick check on the structure of solution sets.

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