b c a b C If def=7, find 2d +a 2e +b 2f+ c ghi g h a b 2d + a 2e+b 2f + c |(Simplify your answer.)
b c a b C If def=7, find 2d +a 2e +b 2f+ c ghi g h a b 2d + a 2e+b 2f + c |(Simplify your answer.)
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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![### Computation of Determinants and Matrix Operations
#### Problem Statement:
Consider a 3x3 matrix \(A\) given by:
\[
A = \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\]
It is known that the determinant of matrix \(A\) equals 7:
\[
\text{det}(A) = \begin{vmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{vmatrix} = 7
\]
#### Task:
You are required to find the determinant of the following matrix:
\[
A' = \begin{pmatrix}
a & b & c \\
2d + a & 2e + b & 2f + c \\
g & h & i
\end{pmatrix}
\]
### Solution:
First, express matrix \(A'\):
\[
A' = \begin{pmatrix}
a & b & c \\
2d + a & 2e + b & 2f + c \\
g & h & i
\end{pmatrix}
\]
To find this determinant, let's denote it by \(\text{det}(A')\).
**Steps to find \(\text{det}(A')\):**
1. We observe that the second row of \(A'\) is a linear combination of the elements of the original second row of \(A\) added to the first row of \(A\).
2. From linear algebra, adding \(\alpha\) times a row to another row does not alter the value of the determinant. This implies if we expand \(A'\) based on the properties of determinants regarding row operations:
\[
\begin{vmatrix}
a & b & c \\
2d + a & 2e + b & 2f + c \\
g & h & i
\end{vmatrix} = \begin{vmatrix}
a & b & c \\
2d & 2e & 2f \\
g & h & i
\end{vmatrix}
\]
3. Factor out the constants (2) from the second row.
Thus,
\[
\text{det}(A') = 2 \times \begin{vmatrix}
a &](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F965f2633-6838-4c2d-b94e-32b85de15479%2F6df2f149-2581-4991-a172-dd5a37f4df44%2Fe8w58to_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Computation of Determinants and Matrix Operations
#### Problem Statement:
Consider a 3x3 matrix \(A\) given by:
\[
A = \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\]
It is known that the determinant of matrix \(A\) equals 7:
\[
\text{det}(A) = \begin{vmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{vmatrix} = 7
\]
#### Task:
You are required to find the determinant of the following matrix:
\[
A' = \begin{pmatrix}
a & b & c \\
2d + a & 2e + b & 2f + c \\
g & h & i
\end{pmatrix}
\]
### Solution:
First, express matrix \(A'\):
\[
A' = \begin{pmatrix}
a & b & c \\
2d + a & 2e + b & 2f + c \\
g & h & i
\end{pmatrix}
\]
To find this determinant, let's denote it by \(\text{det}(A')\).
**Steps to find \(\text{det}(A')\):**
1. We observe that the second row of \(A'\) is a linear combination of the elements of the original second row of \(A\) added to the first row of \(A\).
2. From linear algebra, adding \(\alpha\) times a row to another row does not alter the value of the determinant. This implies if we expand \(A'\) based on the properties of determinants regarding row operations:
\[
\begin{vmatrix}
a & b & c \\
2d + a & 2e + b & 2f + c \\
g & h & i
\end{vmatrix} = \begin{vmatrix}
a & b & c \\
2d & 2e & 2f \\
g & h & i
\end{vmatrix}
\]
3. Factor out the constants (2) from the second row.
Thus,
\[
\text{det}(A') = 2 \times \begin{vmatrix}
a &
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