Q7-58E TRUE OR FALSE58. If聽is an eigen... [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 7: Q7-58E (page 384)

TRUE OR FALSE
58. Ifis an eigenvector of a 2x2matrix $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , then $\overset{鈫�}{v}$ must be an eigenvector of its classical adjoint $A = [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ as well.

Short Answer

Expert verified

The given statement is true.

Step by step solution

Define eigenvector

For any scaler $位$ , if $A \overset{鈫�}{v} = 位 \overset{鈫�}{v}$ , then $\overset{鈫�}{v}$ is the eigenvector of matrix A.

Determine whether the statement is true or false

Let the vector be, $\overset{鈬赌}{v} = [\begin{matrix} v_{1} \\ v_{2} \end{matrix}]$ .

Assume that the vector is an eigenvector of $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ .

Then,

$\begin{array}{rcl} A \overset{鈫�}{v} & = & [\begin{matrix} a v_{1} + b v_{2} \\ c v_{1} + d v_{2} \end{matrix}] \\ = & [\begin{matrix} 位 v_{1} \\ 位 v_{2} \end{matrix}] \end{array}$

Now,

$\begin{array}{rcl} (a d j A) \overset{鈫�}{v} & = & [\begin{matrix} d & - b \\ - c & a \end{matrix}] [\begin{matrix} v_{1} \\ v_{2} \end{matrix}] \\ = & [\begin{matrix} d v_{1} - b v_{2} \\ - c v_{1} + a v_{2} \end{matrix}] \\ = & [\begin{matrix} 位 v_{1} + (a + d) v_{1} \\ 位 v_{2} + (a + d) v_{2} \end{matrix}] \\ = & (位 + a + d) [\begin{matrix} v_{1} \\ v_{2} \end{matrix}] \\ = & (位 + a + d) \overset{鈫�}{v} \end{array}$

So, $\overset{鈫�}{v}$ is also an eigenvalue of A.

Therefore, the given statement is true.

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91影视

TRUE OR FALSE
58. Ifis an eigenvector of a 2x2matrix $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , then $\overset{鈫�}{v}$ must be an eigenvector of its classical adjoint $A = [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ as well.

Short Answer

Step by step solution

Define eigenvector

Determine whether the statement is true or false

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91影视

TRUE OR FALSE58. Ifis an eigenvector of a 2x2matrixA=[abcd], thenv鈫�must be an eigenvector of its classical adjointA=[d-b-ca]as well.

Short Answer

Step by step solution

Define eigenvector

Determine whether the statement is true or false

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Mechanics Maths

Logic and Functions

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

TRUE OR FALSE
58. Ifis an eigenvector of a 2x2matrix $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , then $\overset{鈫�}{v}$ must be an eigenvector of its classical adjoint $A = [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ as well.