If \( A=\begin{bmatrix}1 & 2 & 3\\ -3 & 2 & -1\\ 2 & -4 & 3\end{bmatrix} \), then which of the following is/are correct?
(A) \( A \) is singular matrix
(B) \( |\mathrm{adj}\,A|=1600 \)
(C) \( |4A|=2560 \)
(D) \( (\mathrm{adj}\,A)A=40\,I \) (\( I \) is an identity matrix of order 3)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B), (C) and (D) only
Expanding along the first row:
\( |A|=1(6-4)-2(-9+2)+3(12-4) \)
\( =2+14+24 \)
\( =40 \)
For a matrix of order 3:
\( |\mathrm{adj}\,A|=|A|^{2}=1600 \)
\( |4A|=4^{3}|A|=64\times 40=2560 \)
\( (\mathrm{adj}\,A)A=|A|\,I=40\,I \)
So \( |A|=40 \), \( |\mathrm{adj}\,A|=1600 \), \( |4A|=2560 \) and \( (\mathrm{adj}\,A)A=40\,I \).
Checking the statements
(A) \( A \) is singular matrix — Incorrect. \( |A|=40\ne 0 \).
(B) \( |\mathrm{adj}\,A|=1600 \) — Correct. \( |\mathrm{adj}\,A|=40^{2}=1600 \).
(C) \( |4A|=2560 \) — Correct. \( |4A|=64\times 40=2560 \).
(D) \( (\mathrm{adj}\,A)A=40\,I \) — Correct. \( (\mathrm{adj}\,A)A=|A|\,I=40\,I \).