Question

If A and B are symmetric matrices, then show that AB is symmetric if AB = BA, i.e. A and B commute.

Answer 1

Hint: We know that if the transpose of a matrix is equal to the matrix itself the matrix is known as a symmetric matrix. If we have a symmetric matrix X, then \[{{X}^{T}}=X\]. By using this property of a symmetric matrix we will show the required condition for AB to be symmetric.

Complete step-by-step answer:

We have been given that A and B are symmetric matrices.
Since we know the property of a symmetric matrix that, the transpose of a symmetric matrix is equal to the matrix itself.
So, \[{{A}^{T}}=A\] and \[{{B}^{T}}=B.....(1)\]
Hence the upper case ‘T’ denotes the transpose.
We have been asked to show that AB is symmetric if AB = BA i.e. A and B commute.
Since AB matrix is symmetric, then by using the property of symmetric matrix \[{{\left( AB \right)}^{T}}\] must be equal to AB.
\[\Rightarrow {{\left( AB \right)}^{T}}=AB\]
We know that \[{{\left( {{x}_{1}}{{x}_{2}}{{x}_{3}}.......{{x}_{n}} \right)}^{T}}=\left( {{x}_{n}}^{T}{{x}_{n-1}}^{T}.....{{x}_{2}}^{T}{{x}_{1}}^{T} \right)\]
\[\Rightarrow {{\left( AB \right)}^{T}}={{B}^{T}}{{A}^{T}}\]
Using (1) we get as follows:
\[\begin{align}
  & {{\left( AB \right)}^{T}}=BA \\
 & \Rightarrow {{\left( AB \right)}^{T}}=BA=AB \\
\end{align}\]
The above expression is true if and only if AB = BA.
Therefore, it is shown that if A and B are symmetric matrices then AB is symmetric if and only if AB = BA.

Note: Remember that two matrices A and B are said to be commute matrices if they satisfy the criteria AB = BA. Also remember the property of a matrix that is as follows:
\[{{\left( {{x}_{1}}{{x}_{2}}{{x}_{3}}.......{{x}_{n}} \right)}^{T}}=\left( {{x}_{n}}^{T}{{x}_{n-1}}^{T}.....{{x}_{2}}^{T}{{x}_{1}}^{T} \right)\] which is a very important property to find the transpose of the matrices in multiplication form.