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Found in: Page 93

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Suppose AB = AC, where B and C are $$n \times p$$ matrices and A is invertible. Show that B = C. Is this true, in general, when A is not invertible.

It is proved that $$B = C$$.

See the step by step solution

## Step 1: Condition for an invertible matrix

A $$n \times n$$ matrix A is said to be invertible if there is an equation $$n \times n$$ matrix C such that $$CA = I$$ and $$AC = I$$.

## Step 2: Show that B = C

A matrix that is not invertible is called a singular matrix, and an invertible matrix is called a non-singular matrix.

Multiply both sides of the equation $$AB = AC$$ by $${A^{ - 1}}$$ as shown below:

\begin{aligned}{c}{A^{ - 1}}AB = {A^{ - 1}}AC\\IB = IC\\B = C\end{aligned}

This conclusion is not always true when A is singular.

Hence, it is proved that $$B = C$$.