Question

Let $A$ be a matrix of order $3\times 3$ and det $\left(A\right)=2.$ Then $\text{det}\left(\text{det}\right(A\left)\text{adj}\right(5\text{adj}\left(A^3\right)\left)\right)$ is equal to

• A. $256\times {10}^{11}$
• B. $1024\times {10}^6$
• C. $256\times {10}^6$
• D. $512\times {10}^6$

Answer 1

The correct option is D $512\times {10}^6$

$|A|=2$
$||A|adj\left(5adj{A}^3\right)|$
$=|25|A|adj\left(adj{A}^3\right)|$
$={25}^3|A{|}^3\times |adj{A}^3{|}^2$
$={25}^3\times 2^3\times |A^3{|}^4$
$={25}^3\times 2^3\times 2^{12}=512\times {10}^6$