Question-54
\(\displaystyle A\) and \(\displaystyle B\) are two symmetric matrices. Two square matrices \(\displaystyle P\) and \(\displaystyle Q\) are said to commute if \(\displaystyle PQ=QP\). Which of the following statements are true?
Let us first take the first option where \(\displaystyle A\) and \(\displaystyle B\) commute. This implies that \(\displaystyle AB=BA\). Let us now check for the symmetry of \(\displaystyle AB\):
\[ \begin{equation*} \begin{aligned} ( AB)^{T} & =B^{T} A^{T}\\ & =BA\\ & =AB \end{aligned} \end{equation*} \]
In the second step, we have used the fact that \(\displaystyle A\) and \(\displaystyle B\) are symmetric. In the third step we have used the fact that \(\displaystyle A\) and \(\displaystyle B\) commute. Since \(\displaystyle ( AB)^{T} =AB\), we see that \(\displaystyle AB\) is symmetric. A very similar argument holds for option-(b), which is also true.
Options-(c) requires slightly more work since we have to come up with a counter example. A quick example:
\[ \begin{equation*} A=\begin{bmatrix} 1 & 2\\ 2 & 3 \end{bmatrix} ,\ B=\begin{bmatrix} 3 & 1\\ 1 & 2 \end{bmatrix} \Longrightarrow AB=\begin{bmatrix} 5 & 5\\ 9 & 8 \end{bmatrix} \end{equation*} \]
Option-(d) is clearly false since we can set \(\displaystyle A=B=I\).