If A and B are matrices of same order, then ( A B - B A ) is A

We have matrices A and B of same order. Let P = ( A B - B A ) Then, P = ( A B - B A ) = ( A B ) - ( B A ) = ( B ) (A) - ( A ) B = B A - A B = - ( A B - B A ) = - P Hence, ( A B - B A ) is a skew-symmetric matrix.

Matrices — Class 12 Maths Solution

exemplar objective MCQ NCERT,Exemp,Q.no.63,Page 61

Question

If A and B are matrices of same order, then $\left( {A{B^\prime } - B{A^\prime }} \right)$ is A

(a) skew-symmetric matrix ✓ Correct
(b) null matrix
(c) symmetric matrix
(d) unit matrix

Step-by-step Solution

We have matrices A and B of same order.

Let $P = \left( {A{B^\prime } - B{A^\prime }} \right)$
Then, ${P^\prime } = {\left( {A{B^\prime } - B{A^\prime }} \right)^\prime } = {\left( {A{B^\prime }} \right)^\prime } - {\left( {B{A^\prime }} \right)^\prime }$

$= {\left( {{B^\prime }} \right)^\prime }{(A)^\prime } - {\left( {{A^\prime }} \right)^\prime }{B^\prime } = B{A^\prime } - A{B^\prime }$

$= - \left( {A{B^\prime } - B{A^\prime }} \right) = - P$
Hence, $\left( {A{B^\prime } - B{A^\prime }} \right)$ is a skew-symmetric matrix.

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