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# Prove that if' A and Bare A* I matrices and AB is invertible , then both A and B are invertible* H'arming . You cannot use the formula (LAB ) - !`...

Prove that if A and B are n x n matrices and AB is invertible, then both A and B are invertible.

7 . Prove that if' A and Bare A* I matrices and AB is invertible , then both A and B are invertible*H'arming . You cannot use the formula (LAB ) - !`HI = $-A-, because we are trying to prove in thefirst place that A and & are invertible , so the inverse matrices on the right side of this equation arenot known to exist .Instead , here's a partial hint : Let us give AB a name , say, I."Then , we can find a matrix*"- withthe property that . . . . Stare at what you have and use the Associative Property of MatrixMultiplication . Remember that a left inverse is also a right inverse , and vice - versa , so you onlyneed to show that one equation is true for $1, and another one is true for &. In the process , You!should be able to provide a formula for 1 - and for $ -!`