Use properties of determinants to evaluate the given determinant by inspection. 10 0-2 7 0 06 SOO 5
Use properties of determinants to evaluate the given determinant by inspection. 10 0-2 7 0 06 SOO 5
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Title: Evaluating a Determinant Using Properties**
**Introduction:**
In this section, we demonstrate how to evaluate a determinant using its properties by inspection.
**Matrix Given:**
Below is the 3x3 matrix provided for determinant evaluation:
\[
\begin{vmatrix}
\textcolor{red}{5} & 1 & 0 \\
0 & \textcolor{red}{-2} & 7 \\
0 & 0 & \textcolor{red}{6}
\end{vmatrix}
\]
**Explanation:**
- This is a triangular matrix (specifically, upper triangular as all entries below the main diagonal are zero).
- In such matrices, the determinant is the product of the diagonal elements.
**Calculation:**
To find the determinant, multiply the diagonal elements:
Determinant = \(5 \times (-2) \times 6\)
**Result:**
Determinant = \(-60\)
**Conclusion:**
By using the property that the determinant of a triangular matrix is the product of its diagonal elements, we quickly evaluated this determinant to be \(-60\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F992f5037-f1cc-4483-8aa3-731b184958ad%2F2cfcb9df-56eb-41e9-ac5f-1fe8fa687d64%2Fugz2vp6_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Evaluating a Determinant Using Properties**
**Introduction:**
In this section, we demonstrate how to evaluate a determinant using its properties by inspection.
**Matrix Given:**
Below is the 3x3 matrix provided for determinant evaluation:
\[
\begin{vmatrix}
\textcolor{red}{5} & 1 & 0 \\
0 & \textcolor{red}{-2} & 7 \\
0 & 0 & \textcolor{red}{6}
\end{vmatrix}
\]
**Explanation:**
- This is a triangular matrix (specifically, upper triangular as all entries below the main diagonal are zero).
- In such matrices, the determinant is the product of the diagonal elements.
**Calculation:**
To find the determinant, multiply the diagonal elements:
Determinant = \(5 \times (-2) \times 6\)
**Result:**
Determinant = \(-60\)
**Conclusion:**
By using the property that the determinant of a triangular matrix is the product of its diagonal elements, we quickly evaluated this determinant to be \(-60\).
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