If $A=\begin{bmatrix}2&-1&0\\1&1&2\\-1&0&1\end{bmatrix}$, then which of the following statement(s) is/are correct?
(A) A is singular matrix
(B) $|3A| = 135$
(C) $\text{|adj A|} = 125$
(D) $|A^{-1}|=\frac{1}{5}$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (B) and (D) only
$A = \begin{bmatrix} 2 & -1 & 0 \\ 1 & 1 & 2 \\ -1 & 0 & 1 \end{bmatrix}$
$|A| = 2 \begin{vmatrix} 1 & 2 \\ 0 & 1 \end{vmatrix} - (-1) \begin{vmatrix} 1 & 2 \\ -1 & 1 \end{vmatrix} + 0 \begin{vmatrix} 1 & 1 \\ -1 & 0 \end{vmatrix}$
$|A| = 2(1*1 - 2*0) - (-1)(1*1 - 2*(-1)) + 0 = 2 - (-1)(1+2) = 2 + 3 = 5$
(A) A is singular → $|A| \neq 0$, so Incorrect
(B) $|3A| = 3^3 \cdot |A| = 27 \cdot 5 = 135$ → Correct
(C) $|\text{adj } A| = |A|^{3-1} = |A|^2 = 5^2 = 25$ → Incorrect
(D) $|A^{-1}| = 1/|A| = 1/5$ → Correct