Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

There exists a $2 \times 2$ matrix such that $A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

True, there exist a $2 \times 2$ matrix A such that role="math" localid="1659642050851" $A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Taking the matrix

Suppose the matrix A is.

$A = [\begin{matrix} a & b \\ c & d \end{matrix}]$

Now according to the given equation,

$A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

Step 2: Justification of answer

Put the value of A matrix in the above equation we get.

$[\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

After solving the equation we get

$[\begin{matrix} a + 2 b \\ c + 2 d \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

Now on equating the value of matrix we get

$a + 2 b = 3 c + 2 d = 4$

On solving the equations we get the value of variables

$a = 3 - 2 b c = 4 - 2 d$

Where, are free variables which can have any value. For instance let the value of $b, d = 1$

It implies $a = 1, c = 2$

Thus, we get the matrix

$A = [\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}]$

Hence, it is truethat there exist a $2 \times 2$ matrix A such that $A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

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Linear Algebra With Applications

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Short Answer

There exists a $2 \times 2$ matrix such that $A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

Step by Step Solution

Step 1:Taking the matrix

Step 2: Justification of answer

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There exists a $2 \times 2$ matrix such that $A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$