Question

If A is a square matric $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ then A - $A^T$ is symmetric.

• A. True
• B. False

Answer 1

The correct option is B

False

Here A = $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

$A^T$ will be $\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right]$

$A-A^T$ will be obtained by subtractions of corresponding elements of A^T from A.

$So,A-A^{T}orB=\left[\begin{array}{ccc}(1-1)& (2-4)& (3-7)\\ (4-2)& (5-5)& (6-8)\\ (7-3)& (8-6)& (9-9)\end{array}\right]$

$B=\left[\begin{array}{ccc}0& -2& -4\\ 2& 0& -2\\ 4& 2& 0\end{array}\right]$

Now $B^T$ will be $\left[\begin{array}{ccc}0& 2& 4\\ -2& 0& 2\\ -4& -2& 0\end{array}\right]$

Now we can clearly see that B = -B^T.

So B i.e., $(A–A^T)$is a skew symmetric matrix.

Please note that this is not a particular example where A –$A^T)$comes out to be skew symmetric. It is a properly of a square matrix which says that if A is a square matrix, then A – $A^T)$will always be a skew symmetric matrix.