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Linear Algebra and its Applications
Found in: Page 93
Linear Algebra and its Applications

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

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Short Answer

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 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 \(B = C\).

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