Fsc Part 1 Mathematics (Complete Solution)

#
Q6

If A and B are square matrices of the same order , then explain why in general

**(i) (A+B) ^{2}≠A^{2}+2AB+B^{2}**

Solution:

L.H.S= (A+B)^{2}

=(A+B)(A+B)

=A(A+B)+B(A+B)

=A^{2}+A.B+B.A+B^{2}

As we know that matrix does not has commutative property therfore A.B≠B.A

Hence it is clear that (A+B)^{2}≠A^{2}+2AB+B^{2}

**(ii) (A-B) ^{2}≠A^{2}-2AB+B^{2}**

Solution:

L.H.S= (A-B)^{2}

=(A-B)(A-B)

=A(A-B)-B(A-B)

=A^{2}-A.B-B.A+B^{2}

As we know that matrix does not has commutative property therfore A.B≠B.A

Hence it is clear that (A+B)^{2}≠A^{2}+2AB+B^{2}

**(iii) (A+B)(A-B)≠A ^{2}-B^{2}**

Solution:

L.H.S=(A+B)(A-B)

=A(A-B)+B(A-B)

=A.A-A.B+B.A-B.B

=A^{2}-A.B+B.A-B^{2}

As we know that matrix does not has commutative property therfore A.B≠B.A

Hence it is clear that (A+B)(A-B)≠A^{2}-B^{2}