Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

For the matrix $A = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}],$ write $A = B^{2}$ as discussed in Exercise 30. See Example 1.

$A = B^{2} where, B = \frac{1}{\sqrt{5}} [\begin{matrix} 14 & - 2 \\ - 2 & 11 \end{matrix}]$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information:

$A = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}]$

Step 2: Estimating S:

Consider the matrix below:

$A = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}]$

Now, as shown below, calculate the eigen values:

$\det (A - λl) = 0$

$\Rightarrow |\begin{matrix} 8 - λ & - 2 \\ - 2 & 5 - λ \end{matrix}| = 0$

$\Rightarrow λ^{2} - 13 λ + 36 = 0 \Rightarrow (λ - 4) \cdot (λ - 9) = 0$

As a result, the eigenvalues are 9 and 4, yielding

$D = [\begin{matrix} 9 & 0 \\ 0 & 4 \end{matrix}]$

So,

$D_{1} = [\begin{matrix} 3 & 0 \\ 0 & 2 \end{matrix}]$

The eigen vectors are as follows:

$[\begin{matrix} - 2 \\ 1 \end{matrix}] and [\begin{matrix} 1 \\ 2 \end{matrix}]$

Find S by combining the eigen vectors after they have been converted to unit vectors, as shown below.

$\frac{1}{\sqrt{5}} [\begin{matrix} - 2 \\ 1 \end{matrix}] and \frac{1}{\sqrt{5}} [\begin{matrix} 1 \\ 2 \end{matrix}]$

As a result, we get S, as shown below:

$S = \frac{1}{\sqrt{5}} [\begin{matrix} - 2 & 1 \\ 1 & 2 \end{matrix}]$

Step 3: Estimating B:

Now, as shown below, evaluate B:

$B = {SD}_{1} S$

$= \frac{1}{\sqrt{5}} [\begin{matrix} - 2 & 1 \\ 1 & 2 \end{matrix}] [\begin{matrix} 3 & 0 \\ 0 & 2 \end{matrix}] \frac{1}{\sqrt{5}} [\begin{matrix} - 2 & 1 \\ 1 & 2 \end{matrix}]$

$= \frac{1}{\sqrt{5}} [\begin{matrix} - 2 & 1 \\ 1 & 2 \end{matrix}] [\begin{matrix} 3 & 0 \\ 0 & 2 \end{matrix}] [\begin{matrix} - 2 & 1 \\ 1 & 2 \end{matrix}]$

$= \frac{1}{5} [\begin{matrix} - 2 & 1 \\ 1 & 2 \end{matrix}] [\begin{matrix} - 6 & 3 \\ 2 & 4 \end{matrix}]$

$\Rightarrow B = \frac{1}{5} [\begin{matrix} 14 & - 2 \\ - 2 & 11 \end{matrix}]$

As a result, we can write A as:

$A = B^{2}$

Step 4: Determining the Result:

$A = B^{2} where, B = \frac{1}{5} [\begin{matrix} 14 & - 2 \\ - 2 & 11 \end{matrix}]$

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Linear Algebra With Applications

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Short Answer

For the matrix $A = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}],$ write $A = B^{2}$ as discussed in Exercise 30. See Example 1.

Step by Step Solution

Step 1: Given Information:

Step 2: Estimating S:

Step 3: Estimating B:

Step 4: Determining the Result:

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Linear Algebra With Applications

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Short Answer

For the matrix A=[8-2-25], write A=B2 as discussed in Exercise 30. See Example 1.

Step by Step Solution

Step 1: Given Information:

Step 2: Estimating S:

Step 3: Estimating B:

Step 4: Determining the Result:

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For the matrix $A = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}],$ write $A = B^{2}$ as discussed in Exercise 30. See Example 1.