Question

For the matrix , verify that (i) is a symmetric matrix (ii) is a skew symmetric matrix

Answer 1

(i)

Matrix A is defined as,

A=[ 1 5 6 7 ]

Now, the transpose of the given matrix is,

A ′ =[ 1 6 5 7 ]

Substitute the values in A+ A ′ .

A+ A ′ =[ 1 5 6 7 ]+[ 1 6 5 7 ] =[ 2 11 11 14 ]

For a symmetric matrix, A= A ′ .

So the transpose of A+ A ′ is,

( A+ A ′ ) ′ =[ 2 11 11 14 ] =A+ A ′

Hence, ( A+ A ′ ) is a symmetric matrix.

(ii)

Matrix A is defined as,

A=[ 1 5 6 7 ]

Now, the transpose of the given matrix is,

A ′ =[ 1 6 5 7 ]

Substitute the values in A− A ′ .

A− A ′ =[ 1 5 6 7 ]−[ 1 6 5 7 ] =[ 0 −1 1 0 ]

For a skew-symmetric matrix, A=− A ′ .

So the transpose of A− A ′ is,

( A− A ′ ) ′ =[ 0 1 −1 0 ] =−[ 0 −1 1 0 ] =−( A− A ′ )

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