Question

$IfA=\left[\begin{array}{ccc}2& 1& 3\\ 1& 4& 5\\ 3& 5& 6\end{array}\right]B=\left[\begin{array}{ccc}1& 4& 5\\ 4& 2& 7\\ 5& 8& 3\end{array}\right]$ then which of the following is correct?

• A. Both A and B are symmetric matrices
• B. A is a symmetric matrix and B is not a symmetric matrix
• C. B is a symmetric matrix and A is not a symmetric matrix
• D. Both A and B are not symmetric matrices

Answer 1

The correct option is B

A is a symmetric matrix and B is not a symmetric matrix

We have discussed in the videos the definition for symmetric matrices Remember?

We discussed that it is a square matrix whose transpose is equal to the matrix itself i.e., $A=A^T$ implies that A is a symmetric matrix.

Let’s check the condition for first matrix in the question.

If we take its transpose then all the rows will turn into columns.
So the transpose of A will be,
$A^T=\left[\begin{array}{ccc}2& 1& 3\\ 1& 4& 5\\ 3& 5& 6\end{array}\right]$
Which is same as matrix itself. So A is symmetric.
$B=\left[\begin{array}{ccc}1& 4& 5\\ 4& 2& 7\\ 5& 8& 3\end{array}\right]$

$B^T=\left[\begin{array}{ccc}1& 4& 5\\ 4& 2& 8\\ 5& 7& 3\end{array}\right]\ne B$

So B is not symmetric. Hence option B is correct.