Question

If $A=\begin{bmatrix} 2 & 3 & 0 \\ 1 & sin^{2}x & -cos^{2}x \\ cos^{2}x & cos^{4}x & sin^{4}x \end{bmatrix}$ and $A\left(Adj A\right)=K\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ , then the value of $K$ is

• A. $0$
• B. $-1$
• C. $1$
• D. none of these

Answer 1

Answer: (B)

$A(A d j A)=|A| I_{3}$ $K=|A|=2\left(\sin ^{6} x+\cos ^{6} x\right)-3\left(\sin ^{4} x+\cos ^{4} x\right)$ $=2\left(1-3\sin ^{2} x \cdot \cos ^{2} x\right)-3\left(1-2\sin ^{2} x \cos ^{2} x\right)=-1$