A diagonal matrix is commutative with very square matrix

Question

If $A$ is a diagonal matrix of order $3\times3$ is commutative with every square matrix of order $3\times3$ under multiplication and $trace(A) =12$, then the value of $|A|$ is :

Could someone give me hint to get through this problem

Is $\vert A\vert$ the determinant? – pmichel31415 Apr 19 '16 at 10:58 — pmichel31415, Apr 19 '16 at 10:58

score 4 · Accepted Answer · edited Apr 13 '17 at 12:20

4

Note that commuting with any square matrix implies (for example shown in the answers to this question)$$A=\begin{pmatrix}a&0&0\\0&a&0\\0&0&a\end{pmatrix}.$$

I think you can take it from here.

edited Apr 13 '17 at 12:20

Community

1

answered Apr 19 '16 at 11:01

MooS

31,390

score 2 · Answer 2 · answered Apr 19 '16 at 11:01

2

The only matrices of order $\;\ge2\;$ that commute with any other matrix of the same order are the scalar ones, so

$$A=kI\implies 12=Tr(A)=3k\implies k=4\implies \det A=4^3=64$$

answered Apr 19 '16 at 11:01

DonAntonio

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A diagonal matrix is commutative with very square matrix

2 Answers2