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Suppose $A$ and $B$ are $n \times n, B$ is invertible, and $A B$ is invertible. Show that $A$ is invertible. [Hint: Let $C=A B,$ and solve this equation for $A .$ ]

$A$ is invertible

Algebra

Chapter 2

Matrix Algebra

Section 2

The Inverse of a Matrix

Introduction to Matrices

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

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Suppose $A$ is $n \times n…

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So let's go see a times B and seizing veritable So C minus one exists. So what we do here is we multiply both sides from the right by the inverse of C So we have C see in verse equal a B see embers. But now see, see in verse is by definition of inverse get entity metrics And on the inside we have a me CM birth Now about the associative property or matrix multiplication. We can put the brackets here and this formula is telling us that BC in verse he's the inverse away so easy in vegetable.

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