Choose the correct statement
A. If any two rows or any two columns are identical or proportional, then value of determinant is zero.
B. Minor of an element aij of the determinant of matrix A is the determinant obtained by deleting ith row and jth column
C. If \(A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}\), then A is skew-symmetric matrix
D. If \(A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}\), then \((A + A')\) is symmetric matrix
E. If |A| = 0, then A is non-singular matrix
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A, B, D only
The facts needed:
\(\text{skew-symmetric: } A' = -A, \text{ so every diagonal entry is } 0\)
\(\text{singular: } |A| = 0; \ \text{non-singular: } |A| \ne 0\)
For the matrix in D:
\( A + A' = \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix} + \begin{bmatrix} 1 & 4 \\ 2 & 5 \end{bmatrix} = \begin{bmatrix} 2 & 6 \\ 6 & 10 \end{bmatrix} \)
So the matrix in C has non-zero diagonal entries 1 and 7, \(A + A'\) in D has equal entries 6 and 6 off the diagonal, and |A| = 0 means singular.
Checking the statements
(A) If any two rows or any two columns are identical or proportional, then value of determinant is zero. — Correct. A standard property of determinants.
(B) Minor of an element aij of the determinant of matrix A is the determinant obtained by deleting ith row and jth column — Correct. This is the definition of the minor \(M_{ij}\).
(C) If \(A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}\), then A is skew-symmetric matrix — Incorrect. Its diagonal entries 1 and 7 are not 0 (and 5 ≠ −6).
(D) If \(A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}\), then \((A + A')\) is symmetric matrix — Correct. \(A + A' = \begin{bmatrix} 2 & 6 \\ 6 & 10 \end{bmatrix}\), equal to its transpose.
(E) If |A| = 0, then A is non-singular matrix — Incorrect. |A| = 0 makes A singular.