Question:
If $A$ and $B$ are symmetric matrices, then $A B A$ is
(a) symmetric matrix
(b) skew-symmetric matrix
(c) diagonal matrix
(d) scalar matrix
Solution:
(a) symmetric matrix
Since $A$ and $B$ are symmetric matrices, we get
$A=A^{\prime}$ and $B=B^{\prime}$
$(A B A)^{\prime}=(B A)^{\prime}(A)^{\prime}$
$=A^{\prime} B^{\prime} A^{\prime}$
$=A B A \quad\left[\because A=A^{\prime}\right.$ and $\left.B=B^{\prime}\right]$
Since $(A B A)^{\prime}=A B A, A B A$ is a symmetric matrix.
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