Q.Let $A$ and $B$ be two symmetric matrices. Prove that $AB=BA$ if and only if $AB$ is a symmetric matrix.

Symmetric and Skew-Symmetric Matrices

These are two special kinds of square matrices, defined by how a matrix compares with its own transpose $A'$ (the matrix with rows and columns swapped). They are among the most-tested ideas in the Matrices chapter.

Symmetric matrix

A square matrix $A$ is symmetric if it equals its transpose:

Entries are mirror images across the main diagonal. For example,

Skew-symmetric matrix

A square matrix $A$ is skew-symmetric if its transpose is its negative:

Putting $i=j$ gives $a_{ii} = -a_{ii}$, so $2a_{ii}=0$ — every diagonal entry of a skew-symmetric matrix is $0$. For example,

Key facts

• For any square matrix $A$, the matrix $A + A'$ is always symmetric and $A - A'$ is always skew-symmetric. (Check: $(A+A')' = A' + A = A + A'$.)
• If $A$ is skew-symmetric of odd order, then $\det A = 0$. …