Let 2 A = 9. -1 (a) Show that A 3 (2+3i) 3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Let
\[
A = \begin{bmatrix} 2 & -1 \\ 9 & 2 \end{bmatrix}
\]
(a) Show that
\[
A \begin{bmatrix} i \\ 3 \end{bmatrix} = (2 + 3i) \begin{bmatrix} i \\ 3 \end{bmatrix}.
\]
This task involves verifying the complex multiplication property of matrix \( A \). You are asked to demonstrate how matrix \( A \), when multiplied by a vector, results in the scalar multiplication of that vector by the complex number \( 2 + 3i \).
### Explanation
Given the matrix \( A \):
\[
A = \begin{bmatrix} 2 & -1 \\ 9 & 2 \end{bmatrix}
\]
And the vector:
\[
\begin{bmatrix} i \\ 3 \end{bmatrix}
\]
You need to calculate the product \( A \begin{bmatrix} i \\ 3 \end{bmatrix} \) and show that it equals \( (2 + 3i) \begin{bmatrix} i \\ 3 \end{bmatrix} \).
### Steps
1. Calculate the matrix-vector product:
\[
A \begin{bmatrix} i \\ 3 \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ 9 & 2 \end{bmatrix} \begin{bmatrix} i \\ 3 \end{bmatrix}
\]
2. Verify the result matches:
\[
(2 + 3i) \begin{bmatrix} i \\ 3 \end{bmatrix}
\]
By calculating both sides, you'll confirm the equality and understand the relationship between matrix multiplication and scalar multiplication in the context of complex numbers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F820a86f4-7194-4c38-bb3b-53057670e22e%2F3edda10e-5cbb-4916-ab26-346ccc387ab1%2Frnzmwxb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let
\[
A = \begin{bmatrix} 2 & -1 \\ 9 & 2 \end{bmatrix}
\]
(a) Show that
\[
A \begin{bmatrix} i \\ 3 \end{bmatrix} = (2 + 3i) \begin{bmatrix} i \\ 3 \end{bmatrix}.
\]
This task involves verifying the complex multiplication property of matrix \( A \). You are asked to demonstrate how matrix \( A \), when multiplied by a vector, results in the scalar multiplication of that vector by the complex number \( 2 + 3i \).
### Explanation
Given the matrix \( A \):
\[
A = \begin{bmatrix} 2 & -1 \\ 9 & 2 \end{bmatrix}
\]
And the vector:
\[
\begin{bmatrix} i \\ 3 \end{bmatrix}
\]
You need to calculate the product \( A \begin{bmatrix} i \\ 3 \end{bmatrix} \) and show that it equals \( (2 + 3i) \begin{bmatrix} i \\ 3 \end{bmatrix} \).
### Steps
1. Calculate the matrix-vector product:
\[
A \begin{bmatrix} i \\ 3 \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ 9 & 2 \end{bmatrix} \begin{bmatrix} i \\ 3 \end{bmatrix}
\]
2. Verify the result matches:
\[
(2 + 3i) \begin{bmatrix} i \\ 3 \end{bmatrix}
\]
By calculating both sides, you'll confirm the equality and understand the relationship between matrix multiplication and scalar multiplication in the context of complex numbers.
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