Q23E Matrix聽0100聽is similar to聽000... [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: Q23E (page 164)

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

Short Answer

Expert verified

The above statement is false.

Matrix $[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$ is not similar to $[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$ .

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,

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

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

BP = PA

$鈬� [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$ $鈬� [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$

$鈬� [\begin{matrix} 0 & 0 \\ c & d \end{matrix}] = [\begin{matrix} 0 & a \\ 0 & c \end{matrix}] 鈬� a = 0 c = 0 d = c = 0$

Therefore we get,

$P = [\begin{matrix} 0 & b \\ 0 & 0 \end{matrix}]$

Thus, the matrix $P = [\begin{matrix} 0 & b \\ 0 & 0 \end{matrix}]$ is not invertible matrix.

Hence,

Matrix $[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$ is not similar to $[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$ .

Final Answer

If, $A = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$ and $B = [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$ then there doesn鈥檛 exist any invertible matrix $P = [\begin{matrix} a & b \\ c & d \end{matrix}]$ such that

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

Hence,

Matrix $[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$ is not similar to $[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$ .

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91影视

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

Short Answer

Step by step solution

Definition of similar matrix

Finding an invertible matrix

Final Answer

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91影视

Matrix0100is similar to0001.

Short Answer

Step by step solution

Definition of similar matrix

Finding an invertible matrix

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

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Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

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