a b c a b c If de f=2, find 4d +g 4e + h 4f+i gh i h i a 4d +g 4e +h 4f+i= (Simplify your answer.) g i
a b c a b c If de f=2, find 4d +g 4e + h 4f+i gh i h i a 4d +g 4e +h 4f+i= (Simplify your answer.) g i
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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![### Matrix Equation and Determinant Problem
For this exercise, you are given a matrix equation and asked to find the determinant of a transformed matrix.
#### Given:
\[
\text{If} \begin{vmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{vmatrix} = 2, \ \text{find} \ \begin{vmatrix}
a & b & c \\
4d+g & 4e+h & 4f+i \\
g & h & i
\end{vmatrix}
\]
This is to say, if the determinant of the given matrix is 2, you need to find the determinant of a new matrix formed by modifying the second row of the original matrix.
#### Apply the properties of determinants:
1. **Linearity of Determinant**: The determinant of a matrix is a linear function of the rows (or columns). This means you can separate sums within a row for easier computation.
#### Explanation:
To solve for the determinant of the matrix:
\[ \begin{vmatrix}
a & b & c \\
4d+g & 4e+h & 4f+i \\
g & h & i
\end{vmatrix} \]
Using the linearity property, we can break this determinant down as follows:
\[ \begin{vmatrix}
a & b & c \\
4d+g & 4e+h & 4f+i \\
g & h & i
\end{vmatrix} = \begin{vmatrix}
a & b & c \\
4d & 4e & 4f \\
g & h & i
\end{vmatrix} + \begin{vmatrix}
a & b & c \\
g & h & i \\
g & h & i
\end{vmatrix}
\]
Next, understand the properties of determinants:
- A determinant of a matrix with two identical rows is zero.
- Scaling a row by a constant multiplies the determinant by that constant.
Thus:
\[ \begin{vmatrix}
a & b & c \\
4d & 4e & 4f \\
g & h & i
\end{vmatrix} = 4 \begin{vmatrix}
a & b & c \\
d & e & f \\
g & h &](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F965f2633-6838-4c2d-b94e-32b85de15479%2F2943c993-ca8f-4df4-80f6-4125afdb3a65%2F8yohz6n_processed.png&w=3840&q=75)
Transcribed Image Text:### Matrix Equation and Determinant Problem
For this exercise, you are given a matrix equation and asked to find the determinant of a transformed matrix.
#### Given:
\[
\text{If} \begin{vmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{vmatrix} = 2, \ \text{find} \ \begin{vmatrix}
a & b & c \\
4d+g & 4e+h & 4f+i \\
g & h & i
\end{vmatrix}
\]
This is to say, if the determinant of the given matrix is 2, you need to find the determinant of a new matrix formed by modifying the second row of the original matrix.
#### Apply the properties of determinants:
1. **Linearity of Determinant**: The determinant of a matrix is a linear function of the rows (or columns). This means you can separate sums within a row for easier computation.
#### Explanation:
To solve for the determinant of the matrix:
\[ \begin{vmatrix}
a & b & c \\
4d+g & 4e+h & 4f+i \\
g & h & i
\end{vmatrix} \]
Using the linearity property, we can break this determinant down as follows:
\[ \begin{vmatrix}
a & b & c \\
4d+g & 4e+h & 4f+i \\
g & h & i
\end{vmatrix} = \begin{vmatrix}
a & b & c \\
4d & 4e & 4f \\
g & h & i
\end{vmatrix} + \begin{vmatrix}
a & b & c \\
g & h & i \\
g & h & i
\end{vmatrix}
\]
Next, understand the properties of determinants:
- A determinant of a matrix with two identical rows is zero.
- Scaling a row by a constant multiplies the determinant by that constant.
Thus:
\[ \begin{vmatrix}
a & b & c \\
4d & 4e & 4f \\
g & h & i
\end{vmatrix} = 4 \begin{vmatrix}
a & b & c \\
d & e & f \\
g & h &
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