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  4. If A is diagonal matrix of order 3× 3 is commutative with every square matrix of order 3× 3 under multiplication and trace (A)=12 , then |A| is equal to :-

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Q. If A is diagonal matrix of order $3\times 3$ is commutative with every square matrix of order $3\times 3$ under multiplication and trace $\left(A\right)=12$ , then $\left|A\right|$ is equal to :-

NTA AbhyasNTA Abhyas 2022

Solution:

A diagonal matrix is commutative with every square matrix if it is a scalar matrix i.e. all diagonal elements are equal. So every diagonal element is $4$ and thus $\left|A\right|=4^{3}=64$