68 free MCQs on Matrices with worked answers and explanations. Matrix operations, determinants, inverses, and solving linear systems
Take the timed Matrices chapterwise test →Below are 68 practice questions on Matrices, sorted Easy → Hard. Tap “Show answer & explanation” under any question to check yourself. Want the full theory first? Read the Matrices notes.
Element aij of matrix A sits at the intersection of row i and column j.
Q1.
A matrix with equal number of rows and columns is called a:
Answer: C. Square matrix
Why: A square matrix has the same number of rows and columns, for example a 3x3 matrix.
Q2.
The order of a matrix with 3 rows and 4 columns is:
Answer: B. 3 x 4
Why: Order is written as (rows) x (columns), so a matrix with 3 rows and 4 columns has order 3 x 4.
Q3.
For a 2x2 matrix [[a,b],[c,d]], the determinant is:
Answer: C. ad - bc
Why: det([[a,b],[c,d]]) = ad - bc. Multiply along the main diagonal and subtract the product along the other diagonal.
Q4.
A square matrix in which all elements except the main diagonal are zero is called a:
Answer: D. Diagonal matrix
Why: A diagonal matrix has non-zero elements only on the main diagonal; all off-diagonal elements are zero.
Q5.
The identity matrix of order 3 has main diagonal elements equal to:
Answer: B. 1
Why: The identity matrix I has 1s on the main diagonal and 0s everywhere else. It acts like the number 1 in matrix multiplication.
Q6.
If A is a matrix of order 2 x 3, what is the order of A<sup>T</sup>?
Answer: A. 3 x 2
Why: Transposing swaps rows and columns. If A is 2 x 3, then A<sup>T</sup> is 3 x 2.
Q7.
The transpose of a matrix A is obtained by:
Answer: B. Interchanging rows and columns
Why: The transpose A<sup>T</sup> is formed by converting the rows of A into columns (and columns into rows).
Q8.
A matrix in which all elements are zero is called a:
Answer: B. Zero matrix
Why: A zero matrix or null matrix has every element equal to 0. It is the additive identity for matrix addition.
Q9.
Two matrices can be added only if they have:
Answer: D. The same order (same rows and columns)
Why: Matrix addition is defined only when both matrices have identical orders (same number of rows and columns).
Q10.
If A has order m x n and B has order n x p, then AB has order:
Answer: B. m x p
Why: For AB to exist, columns of A must equal rows of B. The resulting matrix has order m x p.
Q11.
A symmetric matrix satisfies:
Answer: A. A = A<sup>T</sup>
Why: A matrix A is symmetric if A = A<sup>T</sup>, meaning element a<sub>ij</sub> = a<sub>ji</sub> for all i, j.
Q12.
The determinant of the identity matrix I<sub>n</sub> is:
Answer: D. 1
Why: The determinant of any identity matrix equals 1, regardless of its order.
Q13.
A square matrix A is called singular if:
Answer: B. det(A) = 0
Why: A singular matrix has determinant equal to zero and does not have an inverse.
Q14.
If A = [[2,3],[1,4]], what is det(A)?
Answer: B. 5
Why: det(A) = 2 x 4 - 3 x 1 = 8 - 3 = 5.
Q15.
A skew-symmetric matrix satisfies:
Answer: B. A = -A<sup>T</sup>
Why: A is skew-symmetric if A = -A<sup>T</sup>, meaning a<sub>ij</sub> = -a<sub>ji</sub>. All diagonal elements of a skew-symmetric matrix are zero.
Q16.
The number of elements in a matrix of order m x n is:
Answer: D. mn
Why: Total number of elements = m x n (rows times columns). A 3x4 matrix has 12 elements.
Q17.
For any matrix A, (A<sup>T</sup>)<sup>T</sup> equals:
Answer: C. A
Why: Transposing twice returns the original matrix: (A<sup>T</sup>)<sup>T</sup> = A. Transpose is its own inverse.
Q18.
The determinant of a 1 x 1 matrix [a] is:
Answer: B. a
Why: The determinant of a 1x1 matrix [a] is simply the element a itself.
Q19.
A matrix with only one column is called a:
Answer: B. Column matrix
Why: A column matrix (or column vector) has exactly one column and any number of rows.
Q20.
The matrix [[1,0,0],[0,2,0],[0,0,3]] is an example of a:
Answer: D. Diagonal matrix
Why: This is a diagonal matrix because all off-diagonal elements are zero. It is not a scalar matrix because the diagonal values differ.
Q21.
For a 3x3 matrix A, det(2A) =
Answer: D. 8 det(A)
Why: det(kA) = k<sup>n</sup> det(A) for an n x n matrix. For 3x3: det(2A) = 2<sup>3</sup> det(A) = 8 det(A).
Q22.
If A is an n x n matrix, det(A<sup>T</sup>) =
Answer: C. det(A)
Why: The determinant of a matrix equals the determinant of its transpose: det(A<sup>T</sup>) = det(A).
Q23.
The sum of two matrices is defined only when they have the same:
Answer: A. order
Why: Matrix addition requires both matrices to have identical order (dimensions).
Q24.
For the product AB to be defined, the number of columns of A must equal the number of ___ of B:
Answer: A. rows
Why: Matrix multiplication AB needs columns of A to match rows of B.
Q25.
The product of a 2×3 matrix and a 3×2 matrix has order:
Answer: A. 2×2
Why: The product takes the rows of the first and columns of the second: 2×2.
Q26.
A matrix with an equal number of rows and columns is called a ___ matrix:
Answer: A. square
Why: Equal rows and columns define a square matrix.
Q27.
A matrix in which every element is zero is called a ___ matrix:
Answer: A. null
Why: A matrix of all zeros is the null (zero) matrix.
Q28.
The identity matrix has 1s along the ___ and 0s elsewhere:
Answer: A. main diagonal
Why: The identity matrix has 1s on the main diagonal and 0s everywhere else.
Q29.
The transpose of a matrix is obtained by interchanging its:
Answer: A. rows and columns
Why: The transpose Aᵀ swaps rows with columns.
Q30.
If A has order 3×4, then its transpose Aᵀ has order:
Answer: A. 4×3
Why: Transposing swaps the dimensions, so 3×4 becomes 4×3.
Q31.
A square matrix A is called symmetric if:
Answer: A. Aᵀ = A
Why: A symmetric matrix equals its own transpose: Aᵀ = A.
Q32.
A square matrix A is called skew-symmetric if:
Answer: A. Aᵀ = −A
Why: A skew-symmetric matrix satisfies Aᵀ = −A.
Q33.
The diagonal elements of a skew-symmetric matrix are all:
Answer: A. zero
Why: Since Aᵀ = −A, each diagonal element equals its own negative, so it must be zero.
Q34.
Matrix multiplication is, in general:
Answer: A. not commutative
Why: For matrices AB ≠ BA in general, so multiplication is not commutative.
Q35.
A matrix having 3 rows and 5 columns has order:
Answer: A. 3×5
Why: Order is written as (rows)×(columns), so 3×5.
Q36.
The total number of elements in a 3×4 matrix is:
Answer: A. 12
Why: Number of elements = rows × columns = 3 × 4 = 12.
Q37.
Because A + B = B + A for matrices, matrix addition is:
Answer: A. commutative
Why: Matrix addition is commutative (and associative).
Q38.
A diagonal matrix has non-zero entries only on its:
Answer: A. main diagonal
Why: A diagonal matrix has zeros everywhere except possibly on the main diagonal.
Q39.
A scalar matrix is a diagonal matrix whose diagonal elements are all:
Answer: A. equal
Why: A scalar matrix has all its diagonal entries equal (and off-diagonal entries zero).
Q40.
The additive identity for matrices of a given order is the:
Answer: A. null matrix
Why: Adding the null (zero) matrix leaves any matrix unchanged, so it is the additive identity.
Q41.
The eigenvalues of a real symmetric matrix are always:
Answer: B. Real
Why: Proof sketch: if Av = λv with v≠0, take conjugate transpose - v*A = λ̄v*. Then v*Av = λ̄(v*v). But also v*Av = λ(v*v), so λ = λ̄, meaning λ is real. Answer: Real.
Q42.
The Vandermonde determinant for distinct values (a, b, c) equals:
Answer: A. (b-a)(c-a)(c-b)
Why: Vandermonde matrix V = [[1,1,1],[a,b,c],[a²,b²,c²]]. Apply column ops: C<sub>2</sub>−C<sub>1</sub>, C<sub>3</sub>−C<sub>1</sub>, then factor - det = (b−a)(c−a)(c−b). Non-zero iff all values are distinct. Answer: (b−a)(c−a)(c−b).
Q43.
The null space (kernel) of matrix A consists of all vectors x such that:
Answer: B. Ax = 0
Why: The null space (kernel) is defined as Ker(A) = {x : Ax = 0}. By rank-nullity theorem, dim(Ker A) = n − rank(A). It is the solution set of the homogeneous system, always containing the zero vector. Answer: Ax = 0.
Q44.
If A is a 3x3 matrix with rank 2, the null space of A has dimension:
Answer: D. 1
Why: Rank-nullity theorem: rank(A) + nullity(A) = number of columns n. Here 2 + nullity = 3, so nullity = 1. The null space is a 1-dimensional subspace (a line through the origin). Answer: 1.
Q45.
For a Hermitian matrix H, the defining property is:
Answer: D. H = H<sup>dagger</sup> (conjugate transpose)
Why: A Hermitian matrix satisfies H = H<sup>dagger</sup> (H equals its own conjugate transpose). Eigenvalues of a Hermitian matrix are always real.
Q46.
The sum of eigenvalues of matrix A = [[3,1],[2,2]] equals:
Answer: A. 5
Why: Sum of eigenvalues = trace(A) = 3 + 2 = 5. Eigenvalues can also be computed: det(lambda*I-A) = (lambda-3)(lambda-2)-2 = lambda<sup>2</sup>-5lambda+4=0. lambda=1,4. Sum=5.
Q47.
Schur decomposition states that every square matrix A can be written as A = QTQ<sup>-1</sup> where:
Answer: C. T is upper triangular and Q is unitary
Why: Schur: every complex square matrix can be unitarily upper-triangularized: A = QTQ* where Q is unitary and T is upper triangular.
Q48.
A square matrix A is invertible if and only if:
Answer: A. |A| ≠ 0
Why: A matrix has an inverse precisely when its determinant is non-zero (non-singular).
Q49.
The inverse of a matrix A is given by A⁻¹ = (adj A) divided by:
Answer: A. the value |A|
Why: A⁻¹ = adj(A)/|A|, valid when |A| ≠ 0.
Q50.
For invertible matrices A and B, (AB)⁻¹ equals:
Answer: A. B⁻¹A⁻¹
Why: The inverse of a product reverses the order: (AB)⁻¹ = B⁻¹A⁻¹.
Q51.
The transpose of a product, (AB)ᵀ, equals:
Answer: A. BᵀAᵀ
Why: The transpose of a product reverses the order: (AB)ᵀ = BᵀAᵀ.
Q52.
If A is a 2×2 matrix with |A| = 3, then |A⁻¹| equals:
Answer: A. equal to 1/3
Why: |A⁻¹| = 1/|A| = 1/3.
Q53.
A square matrix A satisfying A² = A is called:
Answer: A. idempotent
Why: A matrix equal to its own square is idempotent.
Q54.
A square matrix A satisfying AᵀA = I is called:
Answer: A. orthogonal
Why: An orthogonal matrix satisfies AᵀA = I, so its transpose is its inverse.
Q55.
The trace of a square matrix is the sum of its:
Answer: A. its diagonal elements
Why: The trace is the sum of the entries on the main diagonal.
Q56.
If A and B are both symmetric matrices of the same order, then A + B is:
Answer: A. symmetric
Why: The sum of two symmetric matrices is symmetric, since (A+B)ᵀ = Aᵀ+Bᵀ = A+B.
Q57.
Every square matrix can be written uniquely as the sum of a symmetric matrix and a:
Answer: A. a skew-symmetric one
Why: A = ½(A + Aᵀ) + ½(A − Aᵀ), the sum of a symmetric and a skew-symmetric part.
Q58.
A square matrix A for which Aᵏ = 0 for some positive integer k is called:
Answer: A. nilpotent
Why: A matrix whose some power is the zero matrix is nilpotent.
Q59.
The product of a square matrix and its inverse is the:
Answer: A. identity matrix
Why: By definition, A·A⁻¹ = I, the identity matrix.
Q60.
The number of elements on the main diagonal of a 3×3 matrix is:
Answer: A. 3
Why: An n×n matrix has n elements on its main diagonal, so a 3×3 matrix has 3.
Q61.
If A is a 3×3 matrix with det(A) = 2, then det(2A) equals:
Answer: C. 16
Why: det(kA) = kⁿ·det(A) = 2³·2 = 16 for a 3×3 matrix.
Q62.
A matrix that is both symmetric and skew-symmetric must be:
Answer: B. the null matrix
Why: A = Aᵀ = −A forces A = 0, the null matrix.
Q63.
For a 3×3 matrix A with det(A) = 5, det(adj A) equals:
Answer: C. 25
Why: det(adj A) = det(A)<sup>n−1</sup> = 5² = 25.
Q64.
If a square matrix A satisfies A² = A and A ≠ I, then det(A) equals:
Answer: B. 0
Why: A² = A gives det(A)² = det(A), so det(A) is 0 or 1; since A ≠ I and A is singular here, det(A) = 0.
Q65.
The inverse of the matrix [[2, 3], [1, 2]] is:
Answer: A. [[2, −3], [−1, 2]]
Why: Determinant = 1, so inverse = adjugate = [[2, −3], [−1, 2]].
Q66.
If A is a 3×3 orthogonal matrix, then det(A) equals:
Answer: C. ±1
Why: AAᵀ = I gives det(A)² = 1, so det(A) = ±1.
Q67.
For A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], the trace of AB is:
Answer: C. 16
Why: AB = [[4, 6], [10, 12]], so trace = 4 + 12 = 16.
Q68.
If a square matrix A satisfies A² = 0, then (I − A)⁻¹ equals:
Answer: B. I + A
Why: (I − A)(I + A) = I − A² = I, so (I − A)⁻¹ = I + A.