Q58E If聽is an eigenvector of a 2脳2m... [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: Q58E (page 325)

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

Short Answer

Expert verified

The given statement is false because, in the given matrices $v$ is also an eigenvalue ofrole="math" localid="1668165292169" $a d j A$ .

Step by step solution

Definition of matrices

A matrix (plural matrices) is a square array or desk of numbers, symbols, or expressions organized in rows and columns this is used to symbolize a mathematical item or certainly considered one among its residences in mathematics.

Determine the Matrix

The $m$ rows are horizontal and the $n$ columns are vertical in a $m 脳 n$ matrix.

A variable with subscripts is regularly used to symbolize every detail of a matrix.

If the given matrices,

$v = [\begin{matrix} v_{1} \\ v_{2} \end{matrix}]$

is an eigenvector of

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

Then, the matrices is,

$A v = [\begin{matrix} a v_{1} & b v_{2} \\ c v_{1} & d v_{2} \end{matrix}] = [\begin{matrix} 位 v_{1} \\ 位 v_{2} \end{matrix}]$

Determine the statement is true or false

Now,

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

So, $v$ is also an eigenvalue of $adj A$ .

Therefore, the given statement is True.

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91影视

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

Short Answer

Step by step solution

Definition of matrices

Determine the Matrix

Determine the statement is true or false

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

Ifis an eigenvector of a 2脳2matrix A=[abcd], then v鈫�must be an eigenvector of its classical adjoint adj(A)=[d-b-ca]as well.

Short Answer

Step by step solution

Definition of matrices

Determine the Matrix

Determine the statement is true or false

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Pure Maths

Logic and Functions

Theoretical and Mathematical Physics

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

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