If A is a 3 $\times$ 3 matrix and $|A| = 2$, then $|3\,adj\,(|3A|{A^2})|$ is equal to :
Solution
Given that $A$ is $3 \times 3$ matrix and $|A|=2$
<br/><br/>$$
\begin{aligned}
& \text { Now, | 3adj }\left(|3 A| A^2\right) \text { | } \\\\
& =3^3\left|\operatorname{adj}\left(|3 A| A^2\right)\right| \\\\
& =3^3\left|\operatorname{adj}\left(54 A^2\right)\right| \\\\
& =3^3\left|54 A^2\right|^2 \\\\
& =3^3 \times\left(54^3\right)^2 \times|A|^4 \\\\
& =3^3 \times(54)^6 \times 2^4 ~~~~~ {[|A|=2 \text { given }]} \\\\
& =3^3 \times\left(3^3 \times 2\right)^6 \times 2^4 ~~~~~ {\left[\left(a^m\right)^n=a^{m n}\right]} \\\\
& =3^{11} \times 3^{10} \times 2^{10} ~~~~~~~ {\left[(a b)^m=a^m b^m\right]} \\\\
& =(3)^{11} \times(6)^{10}
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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