Question

How do you prove that AB=BA if and only if AB is also symmetric?

Answer 1

Hint: $AB=BA$ if and only if $AB$ is also symmetric is the given statement which is to be proved. The symmetric matrix is equal to the transpose of the same matrix. To prove the statement we need to use the matrix transpose method. By using transpose the changed matrix will be symmetric. Transpose happens only for the square matrix because both rows and columns will be equal in a square matrix.

Complete step by step answer:
The given statement is $AB=BA\iff$ symmetric which we need to prove
To prove this we need to consider that the matrices $A,B$ are non-null matrices.
But according to the given statement, matrix $A$ is symmetric but the matrix $B$ is not symmetric.
Now, here we need to transpose both matrices $A$ and $B$.
$\Rightarrow AB={\left(AB\right)}^T=B^T{A}^{{}^T}=BA$.
But here
$\Rightarrow A=A^T$
Now, the equation is
$\Rightarrow B^T{A}^T-BA=0$
Here we can take a matrix $A$ as common.
$\Rightarrow (B^T-B)A=0$
$\Rightarrow B^T=B$. This equation is absurd.
So, now the matrix $B$ must be symmetric.
Hence it is proved that $AB=BA$ if and only if $AB$ is also symmetric.

Note:
In the above solution we used transpose to prove the statement. There are many types of symmetric matrices. The numbers in the matrix must be real numbers. The square matrix is also known as the diagonal matrix. Transpose can be applied to any matrix.
But for a matrix to be a symmetric matrix, the matrix must possess a property that is $A\in R$.
If the sum operation and difference operation is done between two symmetric matrices then the resultant matrix will also be symmetric.
If the matrix $A$ has an exponent integer $n$, then the matrix will also be symmetric.