Question

If $A$ is a diagonal matrix of order $3\times 3$ is commutative with every square matrix of order $3\times 3$ under multiplication and trace $\left(A\right)=12$

• A. $\left|A\right|=64$
• B. $\left|A\right|=16$
• C. $\left|A\right|=12$
• D. $\left|A\right|=0$

Answer 1

Answer: (A)

$\left|A\right|=64$

Given $A$ is a diagonal matrix of order $3\times 3$ is commutative with every square matrix of order $3\times 3$ under multiplication.

Then $A$ will be of the form,

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

Given trace of $A$ is $12$.

Then $3a=12$

or, $a=4$.

Now,

$|A|$

$=a^3$

$=4^3$

$=64$.