3 2 Let A = Write 2A. Is det(2A) equal to 2det(A)? 1 9 2A =
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Matrix Scalar Multiplication and Determinant Properties**
**Problem Statement:**
Given a matrix \( A \) defined as:
\[ A = \begin{bmatrix} 3 & 2 \\ 1 & 9 \end{bmatrix} \]
1. Write \( 2A \).
2. Is \( \text{det}(2A) \) equal to \( 2 \cdot \text{det}(A) \)?
**Solution Steps:**
1. **Scalar Multiplication:**
To find \( 2A \), multiply each element of matrix \( A \) by 2.
\[ 2A = 2 \times \begin{bmatrix} 3 & 2 \\ 1 & 9 \end{bmatrix} = \begin{bmatrix} 2 \times 3 & 2 \times 2 \\ 2 \times 1 & 2 \times 9 \end{bmatrix} = \begin{bmatrix} 6 & 4 \\ 2 & 18 \end{bmatrix} \]
Therefore, \( 2A \) is:
\[ \begin{bmatrix} 6 & 4 \\ 2 & 18 \end{bmatrix} \]
2. **Determinant Calculation:**
The determinant of a \( 2 \times 2 \) matrix \( M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by:
\[ \text{det}(M) = ad - bc \]
**First, calculate \( \text{det}(A) \):**
For \( A = \begin{bmatrix} 3 & 2 \\ 1 & 9 \end{bmatrix} \),
\[ \text{det}(A) = (3 \times 9) - (2 \times 1) = 27 - 2 = 25 \]
**Next, calculate \( \text{det}(2A) \):**
For \( 2A = \begin{bmatrix} 6 & 4 \\ 2 & 18 \end{bmatrix} \),
\[ \text{det}(2A) = (6 \times 18) - (4 \times 2) = 108](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F965f2633-6838-4c2d-b94e-32b85de15479%2F2d4d294e-6aa0-4b76-822a-71ceffec350a%2F4a7376_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Scalar Multiplication and Determinant Properties**
**Problem Statement:**
Given a matrix \( A \) defined as:
\[ A = \begin{bmatrix} 3 & 2 \\ 1 & 9 \end{bmatrix} \]
1. Write \( 2A \).
2. Is \( \text{det}(2A) \) equal to \( 2 \cdot \text{det}(A) \)?
**Solution Steps:**
1. **Scalar Multiplication:**
To find \( 2A \), multiply each element of matrix \( A \) by 2.
\[ 2A = 2 \times \begin{bmatrix} 3 & 2 \\ 1 & 9 \end{bmatrix} = \begin{bmatrix} 2 \times 3 & 2 \times 2 \\ 2 \times 1 & 2 \times 9 \end{bmatrix} = \begin{bmatrix} 6 & 4 \\ 2 & 18 \end{bmatrix} \]
Therefore, \( 2A \) is:
\[ \begin{bmatrix} 6 & 4 \\ 2 & 18 \end{bmatrix} \]
2. **Determinant Calculation:**
The determinant of a \( 2 \times 2 \) matrix \( M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by:
\[ \text{det}(M) = ad - bc \]
**First, calculate \( \text{det}(A) \):**
For \( A = \begin{bmatrix} 3 & 2 \\ 1 & 9 \end{bmatrix} \),
\[ \text{det}(A) = (3 \times 9) - (2 \times 1) = 27 - 2 = 25 \]
**Next, calculate \( \text{det}(2A) \):**
For \( 2A = \begin{bmatrix} 6 & 4 \\ 2 & 18 \end{bmatrix} \),
\[ \text{det}(2A) = (6 \times 18) - (4 \times 2) = 108
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