Q11E Find all 2x2聽matrices for which... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q11E (page 307)

Find all 2x2matrices for which $[\begin{matrix} 2 \\ 3 \end{matrix}]$ is an eigenvector with associated eigenvalue -1.

Short Answer

Expert verified

So, the matrix is $\{[\begin{matrix} a \frac{- 2 - 2 a}{3} \\ 3 \frac{- 3 - 2 c}{3} \end{matrix}] Ia, c 鈭� 鈩�\}$ for which $[\begin{matrix} 2 \\ 3 \end{matrix}]$ is an eigenvector with associated eigenvalue -1.

Step by step solution

Step 1: Define the eigenvector

Eigenvector:An eigenvector of Ais a nonzero vector vin $R_{n}$ such that $Av = 位惫$ , for some scalar $位$ .

Solve for A

If $\overset{鈬赌}{v}$ is an Eigen vector of A this means that:

$A \overset{鈬赌}{v} = 位' v$

Now let

A = [\begin{matrix} a & b \\ c & d \end{matrix}], \overset{鈬赌}{v} = and 位 = - 1

and

We want to solve for A.

Evaluation

By the definition of eigenvector,

$A \overset{鈬赌}{v} = 位 \overset{鈬赌}{v} [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 2 \\ 3 \end{matrix}] = - 1 [\begin{matrix} 2 \\ 3 \end{matrix}] [\begin{matrix} 2 a + 3 b \\ 2 c + 3 d \end{matrix}] = [\begin{matrix} - 2 \\ - 3 \end{matrix}] By the equality of matrices, \begin{matrix} 2 a + 3 b = - 2 \\ 2 c + 3 d = - 3 \end{matrix}$

Solving the equations

Solve the equations 2a+3b=-2 and 2c+3d=-3d to get $b = \frac{- 2 - 2 a}{3}$ and $d = \frac{- 3 - 2 c}{3}$ .

Thus, the form of the matrix for which the vector $[\begin{matrix} 2 \\ 3 \end{matrix}]$ is an eigenvector with associated eigenvalue -1 is $[\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} a & \frac{- 2 - 2 a}{3} \\ c & \frac{- 3 - 2 c}{3} \end{matrix}]$ .

Therefore, the required matrices is $\{[\begin{matrix} a & \frac{- 2 - 2 a}{3} \\ c & \frac{- 3 - 2 a}{3} \end{matrix}] Ia, c 鈭� 鈩�\}$

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91影视

Find all 2x2matrices for which $[\begin{matrix} 2 \\ 3 \end{matrix}]$ is an eigenvector with associated eigenvalue -1.

Short Answer

Step by step solution

Step 1: Define the eigenvector

Solve for A

Evaluation

Solving the equations

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91影视

Find all 2x2matrices for which [23]is an eigenvector with associated eigenvalue -1.

Short Answer

Step by step solution

Step 1: Define the eigenvector

Solve for A

Evaluation

Solving the equations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Find all 2x2matrices for which $[\begin{matrix} 2 \\ 3 \end{matrix}]$ is an eigenvector with associated eigenvalue -1.