Question

Define a symmetric matrix. Prove that for

\[A = \begin{bmatrix}2 & 4 \\ 5 & 6\end{bmatrix}\], A + A^T is a symmetric matrix where A^T is the transpose of A.

 

 

Answer 1

A square matrix A is called a symmetric matrix, if `A^T `= A  \[Given: A = \begin{bmatrix}2 & 4 \\ 5 & 6\end{bmatrix} \] 
\[ A^T = \begin{bmatrix}2 & 5 \\ 4 & 6\end{bmatrix}\] 
\[Now, \] 
\[A + A^T = \begin{bmatrix}2 & 4 \\ 5 & 6\end{bmatrix} + \begin{bmatrix}2 & 5 \\ 4 & 6\end{bmatrix}\] 
\[ \Rightarrow A + A^T = \begin{bmatrix}4 & 9 \\ 9 & 12\end{bmatrix} . . . \left( 1 \right)\] 
\[ \left( A + A^T \right)^T = \begin{bmatrix}4 & 9 \\ 9 & 12\end{bmatrix}^T \] 

\[ \left( A + A^T \right)^T = \begin{bmatrix}4 & 9 \\ 9 & 12\end{bmatrix}^T \] 

\[ \left( A + A^T \right)^T = \begin{bmatrix}4 & 9 \\ 9 & 12\end{bmatrix}^T \]   

                 `=  A + A^T  `       [ From eq (1)]

\[ \therefore \left( A + A^T \right)^T = \left( A + A^T \right)\] 

Thus, `( A + A^T )`is a symmetric matrix .