If A and B are symmetric matrices, then ABA is: - Vidyadip

MCQ

If $A$ and $B$ are symmetric matrices, then $\text{ABA}$ is:

✓
Symmetric matrix.
B
Skew$-$symmetric matrix.
C
Diagonal matrix.
D
Scalar matrix.

✓

Answer

Correct option: A.

Symmetric matrix.

Let $\text{A}=\begin{bmatrix}1&2\\2&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}3&2\\2&3\end{bmatrix}$
$\text{AB}=\begin{bmatrix}1&2\\2&1\end{bmatrix}\begin{bmatrix}3&2\\2&3\end{bmatrix}=\begin{bmatrix}7&8\\8&7\end{bmatrix}$
$\text{ABA}=\begin{bmatrix}7&8\\8&7\end{bmatrix}\begin{bmatrix}1&2\\2&1\end{bmatrix}=\begin{bmatrix}23&22\\22&23\end{bmatrix}$

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