Question 12 Marks
If A, B has any two symmetric matrix. Then prove that $A B + B A$ will be a symmetric matrix.
Answer
View full question & answer→ATQ, A and B has any two symmetric matrix.$
\begin{aligned}
\therefore & A^{\prime}=A \text { and } B^{\prime}=B \\
\text { now } & (AB+BA)^{\prime}=(AB)^{\prime}+(BA)^{\prime} \\
& =B^{\prime} A^{\prime}+A^{\prime} B^{\prime} \\
& =BA+AB \\
& {\left[\because B^{\prime}=B \text { and } A^{\prime}=A\right] } \\
& =AB+BA
\end{aligned}
$
$\Rightarrow AB + BA$ has symmetric matrix.
\begin{aligned}
\therefore & A^{\prime}=A \text { and } B^{\prime}=B \\
\text { now } & (AB+BA)^{\prime}=(AB)^{\prime}+(BA)^{\prime} \\
& =B^{\prime} A^{\prime}+A^{\prime} B^{\prime} \\
& =BA+AB \\
& {\left[\because B^{\prime}=B \text { and } A^{\prime}=A\right] } \\
& =AB+BA
\end{aligned}
$
$\Rightarrow AB + BA$ has symmetric matrix.