Does the Trace Commute with Matrix Multiplication? Is $\tr (A B) = \tr (A) \tr (B) $?

Linear algebra problems and solutions

Problem 634

Let $A$ and $B$ be $n \times n$ matrices.

Is it always true that $\tr (A B) = \tr (A) \tr (B) $?

If it is true, prove it. If not, give a counterexample.

 
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Solution.

There are many counterexamples.

For one, take
\[A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \text{ and } B = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}.\]

Then $\tr(A)=1, \tr(B)=1$, and hence $\tr(A) \tr(B) = 1$, while $\tr(AB) = 0$ as $AB = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$.


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