Matrix multiplication is commutative when both matrices are diagonal and have the same dimensions, as well as when they are scalar multiples of each other. The example provided demonstrates that the product of two diagonal matrices, such as \begin{bmatrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{bmatrix} and \begin{bmatrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{bmatrix}, yields the same result regardless of the order of multiplication. This property can be generalized to any dimension, confirming that diagonal matrices commute under multiplication.
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