Which of the following statements are correct?
(A) \(\begin{bmatrix} 3 & -6 \\ -2 & 4 \end{bmatrix}\) is a singular matrix.
(B) \(\begin{bmatrix} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 3 \end{bmatrix}\) is a singular matrix.
(C) \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\) is a singular matrix.
(D) \(\begin{bmatrix} 4 & 1 \\ 1 & 4 \end{bmatrix}\) is a singular matrix.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
A square matrix is singular when its determinant is zero. The four determinants are:
\(\text{(A)}:\ 3 \times 4 - (-6) \times (-2) = 12 - 12 = 0\)
\(\text{(B)}:\ 2\,[(-1) \times 3 - 0 \times 0] = -6\)
\(\text{(C)}:\ 1(45 - 48) - 2(36 - 42) + 3(32 - 35)\)
\(= -3 + 12 - 9 = 0\)
\(\text{(D)}:\ 4 \times 4 - 1 \times 1 = 15\)
So the determinants are 0, −6, 0 and 15 for the matrices in (A), (B), (C) and (D).
Checking the statements
(A) \(\begin{bmatrix} 3 & -6 \\ -2 & 4 \end{bmatrix}\) is a singular matrix. — Correct. Its determinant is 12 − 12 = 0.
(B) \(\begin{bmatrix} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 3 \end{bmatrix}\) is a singular matrix. — Incorrect. Its determinant is 2 × (−1) × 3 = −6.
(C) \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\) is a singular matrix. — Correct. Its determinant is −3 + 12 − 9 = 0.
(D) \(\begin{bmatrix} 4 & 1 \\ 1 & 4 \end{bmatrix}\) is a singular matrix. — Incorrect. Its determinant is 16 − 1 = 15.