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  4. Let M be a square matrix of order 3 whose elements are real numbers and adj(a d j M)=[ 36 0 -4 0 6 0 0 3 6 ] , then the absolute value of Tr(M) is [Here, adjP denotes adjoint matrix of P and Tr(P) denotes trace of matrix P i.e. sum of all principal diagonal elements of matrix P ]

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Q. Let $M$ be a square matrix of order $3$ whose elements are real numbers and
$adj\left(a d j M\right)=\begin{bmatrix} 36 & 0 & -4 \\ 0 & 6 & \, \, 0 \\ 0 & 3 & \, \, 6 \end{bmatrix}$ , then the absolute value of $Tr\left(M\right)$ is
[Here, $adjP$ denotes adjoint matrix of $P$ and $T_{r}\left(P\right)$ denotes trace of matrix $P$ i.e. sum of all principal diagonal elements of matrix $P$ ]

NTA AbhyasNTA Abhyas 2022

Solution:

$adj\left(a d j M\right)=\left|M\right|^{n - 2}M$
$\therefore $ order of $M$ is $3$
$adj\left(a d j M\right)=\left|M\right|M=\begin{bmatrix} 36 & 0 & -4 \\ 0 & 6 & 0 \\ 0 & 6 & 6 \end{bmatrix}$
Taking determinant on both the sides $\left|M\right|^{4}=\begin{bmatrix} 36 & 0 & -4 \\ 0 & 6 & 0 \\ 0 & 3 & 6 \end{bmatrix}$
$\left|M\right|^{4}=6^{4}$
$\Rightarrow $ $\left|M\right|=\pm6$
hence, $M=\begin{bmatrix} \pm6 & 0 & \pm\frac{2}{3} \\ 0 & \pm1 & 0 \\ 0 & \pm\frac{1}{2} & \pm1 \end{bmatrix}$
$\left|T_{r} \left(M\right)\right|=8$