orthogonal, symmetric, skew-symmetric, hermitian, skew Hermitian, normal & unitary matrices and their elementary properties MCQs

1. An orthogonal matrix Q satisfies:

a) Q = Q^T b) Q^TQ = I c) Q = -Q^T d) Q^† = Q

Correct Answer: b) Q^TQ = I

Explanation: Orthogonal matrices satisfy Q^TQ = QQ^T = I.

2. A real symmetric matrix A has:

a) Complex eigenvalues b) Orthogonal eigenvectors c) A = -A^T d) det(A) = 0 always

Correct Answer: b) Orthogonal eigenvectors

Explanation: Real symmetric matrices have real eigenvalues and orthogonal eigenvectors.

3. A real skew-symmetric matrix B satisfies:

a) B = B^T b) B = -B^T c) B^† = B d) B² = I

Correct Answer: b) B = -B^T

Explanation: Skew-symmetric matrices satisfy B^T = -B.

4. A Hermitian matrix H has:

a) H = H^T b) H = -H^† c) H = H^† d) H² = I

Correct Answer: c) H = H^†

Explanation: Hermitian matrices satisfy H = H^† (conjugate transpose).

5. A skew-Hermitian matrix K satisfies:

a) K = K^† b) K = -K^† c) K^T = K d) K² = -I

Correct Answer: b) K = -K^†

Explanation: Skew-Hermitian matrices satisfy K^† = -K.

6. A normal matrix N satisfies:

a) NN^† = N^†N b) N = N^† c) N² = I d) N^T = N

Correct Answer: a) NN^† = N^†N

Explanation: Normal matrices commute with their conjugate transpose.

7. A unitary matrix U satisfies:

a) U^† = U^-1 b) U = U^T c) All elements are real d) det(U) = 1

Correct Answer: a) U^† = U^-1

Explanation: Unitary matrices satisfy U^†U = UU^† = I.

8. The eigenvalues of a Hermitian matrix are:

a) Always real b) Always complex c) Always zero d) Always imaginary

Correct Answer: a) Always real

Explanation: Hermitian matrices have strictly real eigenvalues.

9. Which matrix is both Hermitian and unitary?

a) Identity matrix b) Zero matrix c) Rotation matrix d) All of the above

Correct Answer: a) Identity matrix

Explanation: The identity matrix is both Hermitian (I=I^†) and unitary (I^†=I^-1).

10. The eigenvalues of a skew-Hermitian matrix are:

a) Always real b) Purely imaginary c) Complex with Re=Im d) Always zero

Correct Answer: b) Purely imaginary

Explanation: Skew-Hermitian matrices have eigenvalues that are purely imaginary or zero.

Matrix Algebra MCQs