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Found in: Page 54

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Find a ${\mathbit{n}}{\mathbf{×}}{\mathbit{n}}$ matrix A such that ${\mathbit{A}}\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{=}}{\mathbf{3}}\stackrel{\mathbf{\to }}{\mathbf{x}}$ for all $\stackrel{\mathbf{\to }}{\mathbf{x}}$ in ${{\mathbf{R}}}^{{\mathbf{n}}}$.

Matrix A will be $3I$, where $I$ identity matrix of order is $n×n$.

See the step by step solution

## Step by step Explanation: Step1: System of equation

We have given that $A\stackrel{\to }{x}=3\stackrel{\to }{x}$ for $\stackrel{\to }{x}$ all in ${R}^{n}$.

Which implies that $\stackrel{\to }{x}$ is $n×1$ matrix vector.

## Step2: Linear Transformation

A transformation from ${R}^{m}\to {R}^{n}$ is said to be linear if the following condition holds:

1. Identity of ${R}^{m}$ should be mapped to identity of ${R}^{n}$.
2. $T\left(a+b\right)=T\left(a\right)+T\left(b\right)$For all $a,b\in {R}^{m}$.
3. $T\left(ca\right)=cT\left(a\right)$ Where c is any scalar and $a\in {R}^{m}$.

## Step3: Simplification

We have $A\stackrel{\to }{x}=3\stackrel{\to }{x}$. We can write it as

$A\stackrel{\to }{x}-3\stackrel{\to }{x}=0\phantom{\rule{0ex}{0ex}}\left(A-3\right)\stackrel{\to }{x}=0$

Here, we have taken the vector x is non-zero.

This implies that

$\begin{array}{rcl}A-3& =& 0\\ A& =& 3I\end{array}$

Where $I$is identity matrix of order $n×n$.

Thus, matrix A will be $3I$, where identity matrix of order is $n×n$.