Q. 44.1( 43 Votes )

If A and B are sy

Answer :

To prove: AB – BA is a skew symmetric matrix.

Symmetric matrix: A symmetric matrix is a square matrix that is equal to its transpose. In simple words, matrix A is symmetric if


A = A’


where A’ is the transpose of matrix A.


Skew Symmetric matrix: A skew symmetric matrix is a square matrix that is equal to minus of its transpose. In simple words, matrix A is skew symmetric if


A = -A’


Given: A and B are symmetric matrices i.e.


A = A’ …(1)


B = B’ …(2)


Now calculating the transpose of AB BA,


(AB – BA)’ = (AB)’ – (BA)’


(By property of transpose i.e. (A – B)’ = A’ – B’)


= B’A’ – A’B’


(By property of transpose i.e. (AB)’ = B’A’)


= BA – AB


= -(AB – BA)


Or we can say that: (AB – BA) = - (AB – BA)’


Clearly it satisfies the condition of skew symmetric matrix.


Hence AB BA is a skew symmetric matrix.


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