as we know that a matrix E is said to be symmetric if \(E^T=E\) and skew symmetric if \(E^T=−E.\)

as \(B=A+A^T\) therefore,

\(=B^T =(A+A^T)^T\)

\(=A^T+(A^T)^T\)

\(=A^T+A (As (A^T)^T=A )\)

=B

as \(B^T=B\) therefore, B is symmetric matrix.

hence proved.

As C=A−AT

therefore, \(CT=(A−A^T)^T\)

\(=A^T−(A^T)T\)

\(=A^T−A\)

\(=−(A−A^T)\)

=−C

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,

\(A=1/2(2A)\)

\(=1/2(A+A)\)

\(=1/2(A+A^T+A−A^T)\)

\(=1/2(B+C)\)

where \(B=A+A^T\) and B is a symmetric matrix and \(C=A−A^T\) and C is a skew symmetric matrix.

herefore,

\(A=1/2(B+C)\)

\(=1/2\)(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.

as \(B=A+A^T\) therefore,

\(=B^T =(A+A^T)^T\)

\(=A^T+(A^T)^T\)

\(=A^T+A (As (A^T)^T=A )\)

=B

as \(B^T=B\) therefore, B is symmetric matrix.

hence proved.

As C=A−AT

therefore, \(CT=(A−A^T)^T\)

\(=A^T−(A^T)T\)

\(=A^T−A\)

\(=−(A−A^T)\)

=−C

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,

\(A=1/2(2A)\)

\(=1/2(A+A)\)

\(=1/2(A+A^T+A−A^T)\)

\(=1/2(B+C)\)

where \(B=A+A^T\) and B is a symmetric matrix and \(C=A−A^T\) and C is a skew symmetric matrix.

herefore,

\(A=1/2(B+C)\)

\(=1/2\)(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.