Show that (A + A') is symmetric matrix, if A = (2 4; 3 5).

Show that $(A + A')$ is symmetric matrix, if $A = \begin{pmatrix} 2 & 4 \\ 3 & 5 \end{pmatrix}$.

Given: $A = \begin{pmatrix} 2 & 4 \\ 3 & 5 \end{pmatrix}$

To show: $(A + A')$ is symmetric

Answer

$A + A'$ is symmetric.

1. $A = \begin{pmatrix} 2 & 4\\ 3 & 5\end{pmatrix}$, so $A' = \begin{pmatrix} 2 & 3\\ 4 & 5\end{pmatrix}$ (transpose: rows become columns).
2. $A + A' = \begin{pmatrix} 2+2 & 4+3\\ 3+4 & 5+5\end{pmatrix} = \begin{pmatrix} 4 & 7\\ 7 & 10\end{pmatrix}$ (matrix addition).
3. $(A+A')' = \begin{pmatrix} 4 & 7\\ 7 & 10\end{pmatrix} = A + A'$ (transpose of the sum).
4. Since $(A+A')' = A+A'$, $(A + A')$ is a symmetric matrix (definition of symmetric matrix).