False/True: The matrix True 1 *[23] O False is invertible.
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:**
Determine whether the following statement is true or false:
The matrix \(\begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}\) is invertible.
**Answer Options:**
- True
- False
**Explanation:**
A matrix is invertible if its determinant is non-zero. To find the determinant of a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), use the formula:
\[
\text{det} = ad - bc
\]
For the matrix \(\begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}\), calculate the determinant as follows:
\[
\text{det} = (1)(3) - (3)(2) = 3 - 6 = -3
\]
Since the determinant is \(-3\), which is non-zero, the matrix is invertible. Therefore, the correct answer is "True."](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0f3171b-001d-4e71-b013-5e26782f20a2%2F1a5d74df-c15f-4485-bba5-e8f26bc3852a%2F83h0az7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
Determine whether the following statement is true or false:
The matrix \(\begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}\) is invertible.
**Answer Options:**
- True
- False
**Explanation:**
A matrix is invertible if its determinant is non-zero. To find the determinant of a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), use the formula:
\[
\text{det} = ad - bc
\]
For the matrix \(\begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}\), calculate the determinant as follows:
\[
\text{det} = (1)(3) - (3)(2) = 3 - 6 = -3
\]
Since the determinant is \(-3\), which is non-zero, the matrix is invertible. Therefore, the correct answer is "True."
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