Q15E Matrix聽100-1聽is similar to0110... [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 3: Q15E (page 164)

Matrix $[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ is similar to $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ .

Short Answer

Expert verified

The above statement is true.

Matrix $[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ is similar to $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ .

Step by step solution

Definition of similar matrix

Two square matrices A and B are said to be similar if there exists an invertible matrix P such that

$B = P^{- 1} A P$

Finding an invertible matrix

Let us suppose,

$A = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ and $B = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$

are similar matrices then there exists an invertible matrix $P = [\begin{matrix} a & b \\ c & d \end{matrix}]$ such that

BP = PA

data-custom-editor="chemistry" $鈬� [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] 鈬� [\begin{matrix} c & d \\ a & b \end{matrix}] = [\begin{matrix} a & - b \\ c & - d \end{matrix}] 鈬� a = c d = - b$

Therefore we get,

$P = [\begin{matrix} a & b \\ c & - b \end{matrix}]$

Thus, the matrix $P = [\begin{matrix} a & b \\ c & - b \end{matrix}]$ is an invertible matrix.

Hence,

Matrix $[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ is similar to $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ .

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91影视

Matrix $[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ is similar to $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ .

Short Answer

Step by step solution

Definition of similar matrix

Finding an invertible matrix

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91影视

Matrix100-1is similar to0110.

Short Answer

Step by step solution

Definition of similar matrix

Finding an invertible matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Mechanics Maths

Calculus

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Matrix $[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ is similar to $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ .