[Math] If A and B are symmetric is AB=BA true

linear algebramatrices

Question

if we have matrices $A$ and $B$ which are symmetric. Both of these satisfy:
$$ A=A^T$$
Is $AB=BA$ true ?

Attempt to solve

Now if $A$ and $B$ are symmetric then $AB$ most be symmetric too.

$$ AB=(AB)^T=B^TA^T=BA $$

problem is how do you proof that if $A$ and $B$ are symmetric also $AB$ is symmetric ?

If somone could provide some insight on this that would ge highly appreciated.

Thanks,

Tuki

Best Answer

Let $$A =\begin {bmatrix} a & b\\ b & c \end {bmatrix} \quad\mbox{and}\quad B = \begin {bmatrix} d & e\\ e & f \end {bmatrix}$$ be two symmetric matrices. Then $$AB-BA =\begin {bmatrix} ad+be & ae + bf\\ bd+ec & be + cf\end {bmatrix} -\begin {bmatrix} ad +be & db + ec\\ ea+bf & eb + fc\end {bmatrix}=\begin {bmatrix} 0 & ?\\ ? & 0\end {bmatrix}.$$ Is the right-hand side necessarily the zero matrix?

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