Q46E There exists a non-zero 2 x 2 ma... [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: Q46E (page 164)

There exists a non-zero 2 x 2 matrix A that is similar to 2A.

Short Answer

Expert verified

The above statement is true.

There exists a non-zero 2 x 2 matrix A that is similar to 2A.

Step by step solution

Definition of similar matrix

Let us suppose two square matrices A and B. Then the matrix B is said to be similar to the matrix A if there exists an invertible matrix P such that

$B = P^{- 1} A P A = P^{- 1} (2 A) P$

Finding an invertible matrix

Let us suppose,

$A = [\begin{matrix} 0 & 2 \\ 0 & 0 \end{matrix}] and 2 A = [\begin{matrix} 0 & 4 \\ 0 & 0 \end{matrix}]$

Also, let the matrix Abesimilar matrix to the matrix 2A then there exists an invertible matrix $P = [\begin{matrix} a & b \\ c & d \end{matrix}]$ such that

AP=P(2A)

$鈬� [\begin{matrix} 0 & 2 \\ 0 & 0 \end{matrix}] [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 0 & 4 \\ 0 & 0 \end{matrix}] 鈬� [\begin{matrix} 2 c & 2 d \\ 0 & 0 \end{matrix}] = [\begin{matrix} 0 & 4 a \\ 0 & 4 c \end{matrix}] 鈬� 2 c = 0 鈬� c = 0 2 d = 4 a 鈬� d = 2 a$

Thus, the matrix $P = [\begin{matrix} a & b \\ 0 & 2 a \end{matrix}]$ is an invertible matrix for $a 鈮� 0$ .

There existsaninvertible matrix $P = [\begin{matrix} a & b \\ 0 & 2 a \end{matrix}]$ for $a 鈮� 0$ such that

$A = P^{- 1} (2 A) P$

Final Answer

If $A = [\begin{matrix} 0 & 2 \\ 0 & 0 \end{matrix}] and 2 A = [\begin{matrix} 0 & 4 \\ 0 & 0 \end{matrix}]$ then there exists an invertible matrix $P = [\begin{matrix} a & b \\ 0 & 2 a \end{matrix}]$ for such that

$A = P^{- 1} (2 A) P$

Hence, matrix A is similar to matrix 2A.

So, there exists a non-zero 2 x 2 matrix A that is similar to 2A.

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91影视

There exists a non-zero 2 x 2 matrix A that is similar to 2A.

Short Answer

Step by step solution

Definition of similar matrix

Finding an invertible matrix

Final Answer

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