Question

If A and B are symmetric matrices, then ABA is
(a) symmetric matrix
(b) skew-symmetric matrix
(c) diagonal matrix
(d) scalar matrix

Answer 1

(a) symmetric matrix

$$\begin{gathered} \mathrm{Since}A\mathrm{and}B\mathrm{are}\mathrm{symmetric}\mathrm{matri}ces,\mathrm{we}\mathrm{get} \\ A=A\text{'}\mathrm{and}B=B\text{'} \\ \left(ABA\right)\text{'}=\left(BA\right)\text{'}\left(A\right)\text{'} \\ =A\text{'}B\text{'}A\text{'} \\ =ABA\left[\because A=A\text{'}\mathrm{and}B=B\text{'}\right] \\ \mathrm{Since}\left(ABA\right)\text{'}=ABA,ABA\mathrm{is}\mathrm{a}\mathrm{symmetric}\mathrm{matrix}. \end{gathered}$$