Question

If $A=\left[\stackrel{2}{\underset{0}{-1}}\stackrel{1}{\underset{1}{0}}\stackrel{1}{\underset{3}{2}}\right]$ then prove that

• A. $(A^T{)}^T=A$
• B. $A+A^T$ is a Symmetric Matrix
• C. $A-A^T$ is a Skew-Symmetric Matrix
• D. $A{A}^T$ and $A^{T}A$ are Symmetric Matrices

Answer 1

The correct option is A $(A^T{)}^T=A$

$A=\left(\begin{array}{ccc}2& 1& 1\\ -1& 0& 2\\ 0& 1& 3\end{array}\right)$

$A^T={\left(\begin{array}{ccc}2& 1& 1\\ -1& 0& 2\\ 0& 1& 3\end{array}\right)}^T$=$\left(\begin{array}{ccc}2& -1& 0\\ 1& 0& 1\\ 1& 2& 3\end{array}\right)$

$(A^T{)}^T={\left(\begin{array}{ccc}2& -1& 0\\ 1& 0& 1\\ 1& 2& 3\end{array}\right)}^T=\left(\begin{array}{ccc}2& 1& 1\\ -1& 0& 2\\ 0& 1& 3\end{array}\right)=A$

Hence $(A^T{)}^T=A$