Q18E In the Exercises 17 through 26, ... [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 2: Q18E (page 85)

In the Exercises 17 through 26, find all matrices that commute with the given matrixA.
18. $A = [\begin{matrix} 2 & 3 \\ - 3 & 2 \end{matrix}]$

Short Answer

Expert verified

The matrix $[\begin{matrix} 2 & 3 \\ 鈭� 3 & 2 \end{matrix}]$ is commute with the all matrix of the form. $[\begin{matrix} a & c \\ 鈭� c & a \end{matrix}]$

Step by step solution

Assuming the matrix

Let the given matrix. $A = [\begin{matrix} 2 & 3 \\ 鈭� 3 & 2 \end{matrix}]$

Let the matrix commute with matrix A of the form, $B = [\begin{matrix} a & c \\ b & d \end{matrix}]$ Where $a, b, c, d$ is constant number.

Commutative matrix

Now since multiplication of matrix A and B commute then $A B = B A$ .

$A B = [\begin{matrix} 2 a + 3 b & 2 c + 3 d \\ 鈭� 3 a + 2 b & 鈭� 3 c + 2 d \end{matrix}]$ and $B A = [\begin{matrix} 2 a 鈭� 3 c & 3 a + 2 c \\ 2 b 鈭� 3 d & 3 b + 2 d \end{matrix}]$

Now

$\begin{array}{l} A B = B A \\ 鈬� [\begin{matrix} 2 a + 3 b & 2 c + 3 d \\ 鈭� 3 a + 2 b & 鈭� 3 c + 2 d \end{matrix}] = [\begin{matrix} 2 a 鈭� 3 c & 3 a + 2 c \\ 2 b 鈭� 3 d & 3 b + 2 d \end{matrix}] \end{array}$

On equating the matrix we get equations

$\begin{matrix} 2 a + 3 d = 2 a 鈭� 3 c \\ 2 c + 3 d = 3 a + 2 d \\ 鈭� 3 a + 2 b = 2 b 鈭� 3 d \\ 鈭� 3 c + 2 d = 3 b + 2 d \end{matrix}$

On solving these equations, we get

$a = a, b = 鈭� c, c = c, d = a$

Hence,the matrix $[\begin{matrix} 2 & 3 \\ 鈭� 3 & 2 \end{matrix}]$ is commute with the all matrix of the form. $[\begin{matrix} a & c \\ 鈭� c & a \end{matrix}]$

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91影视

In the Exercises 17 through 26, find all matrices that commute with the given matrixA.
18. $A = [\begin{matrix} 2 & 3 \\ - 3 & 2 \end{matrix}]$

Short Answer

Step by step solution

Assuming the matrix

Commutative matrix

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91影视

In the Exercises 17 through 26, find all matrices that commute with the given matrixA.18.A=[23-32]

Short Answer

Step by step solution

Assuming the matrix

Commutative matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Study anywhere. Anytime. Across all devices.

In the Exercises 17 through 26, find all matrices that commute with the given matrixA.
18. $A = [\begin{matrix} 2 & 3 \\ - 3 & 2 \end{matrix}]$