Question

If A is a diagonal matrix of order $3\times 3$ is a commutative with every square matrix of order $3\times 3$ under multiplication and tr (A) = 12, then the value of $\mid A\mid$ is

Answer 1

Consider the problem

Any $3\times 3$ diagonal matrix which commutes with all other $3\times 3$ square matrices is

$A=\left[\begin{array}{ccc}x& 0& 0\\ 0& x& 0\\ 0& 0& x\end{array}\right]$

$Trace=sumofdiagonalentries=12$

$\begin{array}{c}\Rightarrow a+a+a=12\\ \Rightarrow 3a=12\\ \Rightarrow a=4\end{array}$

$\Rightarrow A=\left[\begin{array}{ccc}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right]$

Expand along $R_1$

$\begin{array}{c}\Rightarrow \left|A\right|=4\left(4\times 4-0\right)\\ =4\times 16\\ =64\end{array}$