If \( A \) is a square matrix of order 3 such that \( A\,(\mathrm{adj}\,A)=\begin{bmatrix}-3 & 0 & 0\\0 & -3 & 0\\0 & 0 & -3\end{bmatrix} \) , then which of the following is NOT correct?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( A \) is a singular matrix.
For any square matrix, \( A\,(\mathrm{adj}\,A)=|A|\,I \). Comparing with the given matrix, which is \( -3I \):
\( \displaystyle |A|=-3 \)
Option (1) \( A \) is an invertible matrix. — Correct. \( |A|=-3\ne 0 \).
Option (2) \( |\mathrm{adj}\,A|=9 \) — Correct. \( |\mathrm{adj}\,A|=|A|^{n-1}=(-3)^2=9 \).
Option (3) \( A^{-1} \) is a non-singular matrix. — Correct. \( |A^{-1}|=\frac{1}{|A|}=-\frac{1}{3}\ne 0 \).
Option (4) \( A \) is a singular matrix. — Incorrect. A singular matrix has determinant 0, but \( |A|=-3 \).
The question asks for the statement that is NOT correct, so the answer is Option (4), the only statement that is not true.