Question

If ‘A’ is a symmetric matrix then prove that B’AB is a symmetric.

Answer 1

Hint: o check whether a matrix is symmetric or not just take its transpose if the matrix obtained after transpose is identical with the given matrix then it will be named symmetric otherwise not.
We use the following formula For symmetric matrix (A)’ = (A)

Complete step-by-step answer:
In this problem it is given that matrix A is a symmetric matrix.
We know that a matrix is said to be a symmetric matrix if the matrix obtained after the transpose of the given matrix is the same as that of the taken matrix.
Therefore, for a symmetric matrix we can write (A)’= A.
Hence, to check if any matrix is a symmetric matrix we just take its transpose and see if we get the same matrix as a result after transpose. Then, it will be called symmetric and if not then the given matrix will not be symmetric.
Now, to check whether B’AB is a symmetric matrix we will take its transpose.
Considering (B’AB)’
 $
   = (B)'(A)'(B')' \\
   = B'(A)'B\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {\because \,\,(B')' = B} \right\} \\
   = B'AB\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\because \,\,A\,\,is\,\,given\,\,symmetric) \\
  $
Hence, from above we see that the transpose of B’AB is the same matrix B’AB.
Therefore, we can say that B’AB is a symmetric matrix.

Note: On taking transpose of a given matrix either rows are changed to columns or columns are changed to rows. But in case of symmetric matrix rows and columns are such that even after taking the transpose of the matrix it results as an identical matrix as given. Hence, to discuss whether a matrix is symmetric or not it is just enough to take its transpose.