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If a matrix A has inverses B and C, then which one of the following is correct?

If a matrix A has inverses B and C, then which one of the following is correct? Correct Answer B should be equal to C

Concept:

If A, B, C are matrices then A(BC) = (AB)C

A-1 A = I

Calculations:

Given that, A has inverse B

⇒ AB = I = BA ....(1)

Also, A has inverse C.

⇒ AC = I = CA ....(2) 

From (1) and (2),

AB = AC ....(3)

Pre multiply equation (3) by A-1

⇒A-1 (AB) = A-1 (AC)

⇒A-1 (AB) = A-1 (AC)

⇒(A-1 A)B = (A-1A)C

⇒(IB) = (IC)

⇒ B = C.

Hence, if  a matrix A has inverses B and C,then B should be equal to C

 

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