Question

For the matrix A=\(\begin{bmatrix}1&5\\6&7\end{bmatrix}\),verify that 
  I. (A+A') is a symmetric matrix
  II. (A-A') is a skew symmetric matrix

Answer 1

A'= \(\begin{bmatrix}1&6\\5&7\end{bmatrix}\)

(i)A+A'=\(\begin{bmatrix}1&5\\6&7\end{bmatrix}\)+\(\begin{bmatrix}1&6\\5&7\end{bmatrix}\)

=\(\begin{bmatrix}2&11\\11&14\end{bmatrix}\)

therefore (A+A')'= \(\begin{bmatrix}2&11\\11&14\end{bmatrix}\)=A+A'
Hence,(A+A') is a symmetric matrix. 

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(ii)A-A'= \(\begin{bmatrix}1&5\\6&7\end{bmatrix}\)-\(\begin{bmatrix}1&6\\5&7\end{bmatrix}\)

=\(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\)

(A-A')'=\(\begin{bmatrix}0&1\\-1&0\end{bmatrix}\)=-\(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\)=-(A-A')

Hence,(A-A') is a skew-symmetric matrix.