Note: for each question below, enter a numerical value only. Let's find the following determinant by cofactor expansion along the third row: 4 3 -7 10 -1 -5 = (-2) X-(-3) - Y+7 Z= ..., -2 -3 7 where the values of the respective 2 X 2 determinants in the expansion are: 0.0.0 = A/ Z N 4 and, finally, the value of the original determinant is 10 -2 A 3 -7 -1 -5 = ... -3 7
Note: for each question below, enter a numerical value only. Let's find the following determinant by cofactor expansion along the third row: 4 3 -7 10 -1 -5 = (-2) X-(-3) - Y+7 Z= ..., -2 -3 7 where the values of the respective 2 X 2 determinants in the expansion are: 0.0.0 = A/ Z N 4 and, finally, the value of the original determinant is 10 -2 A 3 -7 -1 -5 = ... -3 7
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
Help solve this practice problem, please be clear/detailed
![**Determinant Calculation Using Cofactor Expansion**
**Objective:**
Calculate the determinant of a \(3 \times 3\) matrix by cofactor expansion along the third row.
**Given Matrix:**
\[
\begin{vmatrix}
4 & 3 & -7 \\
10 & -1 & -5 \\
-2 & -3 & 7 \\
\end{vmatrix}
\]
**Cofactor Expansion:**
\[
= (-2) \cdot X - (-3) \cdot Y + 7 \cdot Z = \ldots
\]
where the values of the respective \(2 \times 2\) determinants in the expansion are:
- **X** (determinant excluding the third row and first column)
- **Y** (determinant excluding the third row and second column)
- **Z** (determinant excluding the third row and third column)
**Calculation Boxes:**
- \(X = \) \(\boxed{\hphantom{00}}\)
- \(Y = \) \(\boxed{\hphantom{00}}\)
- \(Z = \) \(\boxed{\hphantom{00}}\)
**Final Determinant Calculation:**
\[
= \begin{vmatrix}
4 & 3 & -7 \\
10 & -1 & -5 \\
-2 & -3 & 7 \\
\end{vmatrix}
= \ldots
\]
- \(\boxed{\hphantom{00}}\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff1b6656-acf4-4e95-902d-43b1d88f0837%2F2c0e9da8-bc77-4dae-8721-620fcc7407f0%2Fdsottzs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determinant Calculation Using Cofactor Expansion**
**Objective:**
Calculate the determinant of a \(3 \times 3\) matrix by cofactor expansion along the third row.
**Given Matrix:**
\[
\begin{vmatrix}
4 & 3 & -7 \\
10 & -1 & -5 \\
-2 & -3 & 7 \\
\end{vmatrix}
\]
**Cofactor Expansion:**
\[
= (-2) \cdot X - (-3) \cdot Y + 7 \cdot Z = \ldots
\]
where the values of the respective \(2 \times 2\) determinants in the expansion are:
- **X** (determinant excluding the third row and first column)
- **Y** (determinant excluding the third row and second column)
- **Z** (determinant excluding the third row and third column)
**Calculation Boxes:**
- \(X = \) \(\boxed{\hphantom{00}}\)
- \(Y = \) \(\boxed{\hphantom{00}}\)
- \(Z = \) \(\boxed{\hphantom{00}}\)
**Final Determinant Calculation:**
\[
= \begin{vmatrix}
4 & 3 & -7 \\
10 & -1 & -5 \\
-2 & -3 & 7 \\
\end{vmatrix}
= \ldots
\]
- \(\boxed{\hphantom{00}}\)
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