Q24E In Exercises 17 through 26, find... [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: Q24E (page 85)

In Exercises 17 through 26, find all matrices that commute with the given matrixA.
$A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}]$

Short Answer

Expert verified

The matrix $A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}]$ is commute with all the matrices of the form $[\begin{matrix} a & d & 0 \\ b & e & 0 \\ 0 & 0 & i \end{matrix}]$

Step by step solution

Assuming the matrix

Let the given matrix

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

Let the matrix commute with matrix A of the form $B = [\begin{matrix} a & d & g \\ b & e & h \\ c & f & i \end{matrix}]$ , where $a, b, c, d$ is constants.

Commutative matrix

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

$A B = [\begin{matrix} 2 a & 2 d & 2 g \\ 2 b & 2 e & 2 h \\ 2 c & 2 f & 2 i \end{matrix}], B A = [\begin{matrix} 2 a & 2 d & 2 g \\ 2 b & 2 e & 2 h \\ 2 c & 2 f & 2 i \end{matrix}]$

Now

$A B = B A A B = [\begin{matrix} 2 a & 2 d & 2 g \\ 2 b & 2 e & 2 h \\ 2 c & 2 f & 2 i \end{matrix}], B A = [\begin{matrix} 2 a & 2 d & 2 g \\ 2 b & 2 e & 2 h \\ 2 c & 2 f & 2 i \end{matrix}]$

On equating the corresponding elements of obtained matrix, we get equations

$2 a = 2 a 2 d = 2 d 2 g = 3 g 2 b = 2 b 2 e = 2 e 2 h = 3 h 2 c = 3 c 3 f = 2 f 3 i = 3 i$

On solving these equations, we get

$a = a, b = b, c = 0, d = d, e = e, f = 0, g = 0, h = 0, i = i$

Hence, Matrix $A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}]$ is commute with all the matrix of the form $[\begin{matrix} a & d & 0 \\ b & e & 0 \\ 0 & 0 & i \end{matrix}]$

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91影视

In Exercises 17 through 26, find all matrices that commute with the given matrixA.
$A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}]$

Short Answer

Step by step solution

Assuming the matrix

Commutative matrix

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91影视

In Exercises 17 through 26, find all matrices that commute with the given matrixA.A=200020003

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

Statistics

Pure Maths

Logic and Functions

Geometry

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Exercises 17 through 26, find all matrices that commute with the given matrixA.
$A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}]$