Question

Let $M$ and $N$ are two non singular matrices of order $3$ with real entries such that $\left(a d j M\right)=2N$ and $\left(a d j N\right)=M$ . If $MN=\lambda Ι$ , then the value of $\lambda $ is equal to (where, $\left(a d j X\right)$ represents the adjoint matrix of matrix $X$ and $I$ represents an identity matrix)

Answer 1

$\left(a d j M\right)=2N\Rightarrow \left|a d j \left(M\right)\right|=\left|2 N\right|\Rightarrow \left(\left|M\right|\right)^{2}=8\left|N\right|...\left(1\right)$
$\left(a d j N\right)=M\Rightarrow \left|a d j \left(N\right)\right|=\left|M\right|\Rightarrow \left(\left|N\right|\right)^{2}=\left|M\right|...\left(2\right)$
From $\left(1\right)$ and $\left(2\right)$ , we get,
$\left|N\right|=2,\left|M\right|=4$
Now, $MN=\left(a d j N\right)N=\left|N\right|Ι=2Ι\Rightarrow \lambda =2$