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Linear Algebra and its Applications

Found in: Page 93

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

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 \(B = C\).

Most popular questions for Math Textbooks

Suppose A is \(n \times n\) and the equation \(A{\bf{x}} = {\bf{0}}\) has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to \({I_n}\). By Theorem 7, this shows that A must be invertible. (This exercise and Exercise 24 will be cited in Section 2.3.)

2. Find the inverse of the matrix \(\left( {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{\bf{7}}&{\bf{4}}\end{aligned}} \right)\).

Suppose A is a \(3 \times n\) matrix whose columns span \({\mathbb{R}^3}\). Explain how to construct an \(n \times 3\) matrix D such that \(AD = {I_3}\).

Solve the equation \(AB = BC\) for A, assuming that A, B, and C are square and B is invertible.

Let \(A = \left( {\begin{aligned}{*{20}{c}}1&1&1\\1&2&3\\1&4&5\end{aligned}} \right)\), and \(D = \left( {\begin{aligned}{*{20}{c}}2&0&0\\0&3&0\\0&0&5\end{aligned}} \right)\). Compute \(AD\) and \(DA\). Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a \(3 \times 3\) matrix B, not the identity matrix or the zero matrix, such that \(AB = BA\).

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