If $A$ is a square matrix such that $A(\operatorname{adj} A)=\left[\begin{array}{lll}4 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 4\end{array} ight]$, then $\operatorname{det}(\operatorname{adj} A)$ is equal to

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If $A$ is a square matrix such that $A(\operatorname{adj} A)=\left[\begin{array}{lll}4 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 4\end{array}ight]$, then $\operatorname{det}(\operatorname{adj} A)$ is equal to

Aicharya · Accepted Answer

$\begin{aligned} & 	ext { Given } A(\operatorname{adj} A)=\left[\begin{array}{lll}4 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 4\end{array}ight] \ & 	ext { Now, }|A(\operatorname{adj} A)|=\left|\begin{array}{lll}4 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 4\end{array}ight| \ & \Rightarrow|A||\operatorname{adj} A|=4^3 \ & \because \quad|\operatorname{adj} A|=|A|^{n-1} \ & 	ext { Here, } \quad n=3 \ & 	herefore \quad|\operatorname{adj} A|=|A|^2 \ & \left.\Rightarrow \quad|A| A Aight|^2=64 \ & \Rightarrow \quad|A|^3=(4)^3 \ & \Rightarrow \quad|A|=4 \ & \end{aligned}$ $\begin{aligned} & 	ext { Now, } \quad|\operatorname{adj} A|=|A|^2=(4)^2 \ & 	herefore \quad|\operatorname{adj} A|=16 \ & \end{aligned}$

If $A$ is a square matrix such that $A(\operatorname{adj} A)=\left[\begin{array}{lll}4 & 0 & 0 \\ 0 & 4 & 0…

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