If A is a 3 [] 3 matrix and |A| = 2, then [|3,adj,(|3A|A^2)|] is equal to :

Question

If A is a 3 $$\times$$ 3 matrix and $|A| = 2$, then $$|3\,adj\,(|3A|{A^2})|$$ is equal to :

Accepted Answer

Given that $A$ is $3 \times 3$ matrix and $|A|=2$

$$
\begin{aligned}
& \text { Now,, 3adj }\left(|3 A| A^2\right) \text {, } \\
& =3^3\left|\operatorname{adj}\left(|3 A| A^2\right)\right| \\
& =3^3\left|\operatorname{adj}\left(54 A^2\right)\right| \\
& =3^3\left|54 A^2\right|^2 \\
& =3^3 \times\left(54^3\right)^2 \times|A|^4 \\
& =3^3 \times(54)^6 \times 2^4 ~~~~~ {[|A|=2 \text { given }]} \\
& =3^3 \times\left(3^3 \times 2\right)^6 \times 2^4 ~~~~~ {\left[\left(a^m\right)^n=a^{m n}\right]} \\
& =3^{11} \times 3^{10} \times 2^{10} ~~~~~~~ {\left[(a b)^m=a^m b^m\right]} \\
& =(3)^{11} \times(6)^{10}
\end{aligned}
$$

Solutions · 1