Fsc Part 1 Mathematics (Complete Solution)

Q17
If A and B are non-singular matrices , then show that

(i) (AB)-1=B-1 A-1

SOLUTION:

Let (AB)(B-1A-1)

=A(BB-1)A-1=A I A-1=I

Now (B-1A-1)(AB)

=B-1(A-1A)B=B-1I B=B-1B=I

Thus (AB)(B-1A-1)=(B-1A-1)(AB)=I

It shows that AB is the inverse ho B-1A-1

=> (AB)-1=B-1A-1



(ii) (A-1)-1=A

SOLUTION:

Since A-1A=I

and AA-1=I

It shows that A-1 be the inverse of A

(A-1)-1=A

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