If A and B are symmetric matrices, then ABA is


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


(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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