Question:
If A and B are symmetric matrices of the same order, write whether AB − BA is symmetric or skew-symmetric or neither of the two.
Solution:
Since $A$ and $B$ are symmetric matrices, $A^{T}=A$ and $B^{T}=B$.
Here,
$(A B-B A)^{T}=(A B)^{T}-(B A)^{T}$
$\Rightarrow(A B-B A)^{T}=B^{T} A^{T}-A^{T} B^{T} \quad\left[\because(A B)^{T}=B^{T} A^{T}\right]$
$\Rightarrow(A B-B A)^{T}=B A-A B \quad\left[\because B^{T}=B\right.$ and $\left.A^{T}=A\right]$
$\Rightarrow(A B-B A)^{T}=-(A B-B A)$
Therefore, $A B-B A$ is skew-symmetric.
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