Let A and B be two symmetric matrices of order 3.

Question

Let $A$ and $B$ be two symmetric matrices of order 3.

Statement-1: $A(BA)$ and $(AB)A$ are symmetric matrices.

Statement-2: $AB$ is a symmetric matrix if the matrix multiplication of $A$ with $B$ is commutative.

Accepted Answer

Since $A^T=A,\ B^T=B$: $\big(A(BA)\big)^T=(BA)^TA^T=(A^TB^T)A^T=(AB)A=A(BA)$ (using associativity and $A^T=A,B^T=B$), so $A(BA)$ is symmetric. Similarly $(AB)A$ is symmetric. Statement-1 is true.

If $AB=BA$: $(AB)^T=B^TA^T=BA=AB$, so $AB$ is symmetric --- Statement-2 is also true as a standalone fact, but it does not directly explain why $A(BA)$ and $(AB)A$ are always symmetric (that holds regardless of whether $AB=BA$), so Statement-2 is not the correct explanation of Statement-1.

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