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Expert-verifiedFound in: Page 93

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 \(B = C\).

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 \(AC = I\).

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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