as \(B=A+A^T\) therefore,
\(=A^T+A (As (A^T)^T=A )\)
as \(B^T=B\) therefore, B is symmetric matrix.
as \(C^T=−C\) therefore, C is skew symmetric matrix. Hence proved.
Now we have to show that every nxxn matrix can be expressed as the sum of the symmetric and skew symmetric matrix. Let A be nxxn matrix. therefore,
where \(B=A+A^T\) and B is a symmetric matrix and \(C=A−A^T\) and C is a skew symmetric matrix.
\(=1/2\)(symmetric matrix +skew symmetric matrix)
therefore, it has been showed that any matrix A of order nxxn can be expressed as the sum of symmetric and skew symmetric matrix.