(7) If A and B are a square matrix of the same order. Prove that (ABA^−1)^3=AB^3A^−1(8) If A, B and A+B are aach norsingular. Prove thatA(A+B)^-1B=B(A+B)^-1A=(A^-1+B^-1)^-1

Suman Cole

Suman Cole

Answered question

2021-02-21

(7) If A and B are a square matrix of the same order. Prove that (ABA1)3=AB3A1

Answer & Explanation

Gennenzip

Gennenzip

Skilled2021-02-22Added 96 answers

Let A and B are n*n matrix. Then we have
(ABA1)3=(ABA1)(ABA1)(AB1)=(AB)(A1A)B(A1A)(BA1) Matrix multiplication associative
=(AB)B(BA1)
AB3A1
Let A, B and A + B are aach norsingular. We have to show A(A+B)1B=B(A+B)1A=(A1+B1)1
Note that we need to show that A(A+B)1B is inverse of (A1+B1).
So, our first task is to show A(A+B)1B(A1+B1)=(A1+B1)A(A+B)1B=1

Notice that
A(A+B)1B(A1+B1)=A(A+B)1(BA1+1)=A(A+B)1(A+B)A1=I
Also we have
(A1+B1)A(A+B)1B=(I+B1A)(A+B)1B=B1(B+A)(A+B)
This proves that A(A+B)1B is inverse of (A1+B1)
Similarly, we can prove that B(A+B)1A is inverse of (A1+B1)

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