### Matrix Determinants and Operations **Problem Statement:** Suppose that the matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) has determinant \( \text{det}(A) = -3 \). And the matrix \( B = \begin{bmatrix} x & b & c \\ y & e & f \\ z & h & i \end{bmatrix} \) has determinant \( \text{det}(B) = 2 \). **Objective:** 1. Find the determinant of the following matrix: \[ \text{det} \left( \begin{bmatrix} a + 5x & c & b \\ d + 5y & f & e \\ g + 5z & j & h \end{bmatrix} \right) \] 2. Find the determinant of \( -2A^2B^{-1} \). **Details for Explanation:** 1. The matrix inside the determinant in the first problem has elements adjusted by adding 5 times the corresponding elements of another matrix. 2. For the second problem, the operation involves finding \( A^2 \) (the square of matrix \( A \)), multiplying it by \(-2\), and then multiplying by the inverse of matrix \( B \) (denoted \( B^{-1} \)). --- ### Solution Approach: 1. **Determinant of the Sum Adjusted Matrix:** The given matrix is: \[ \begin{bmatrix} a + 5x & c & b \\ d + 5y & f & e \\ g + 5z & j & h \end{bmatrix} \] To solve this, we need to apply properties of determinants, row operations and possibly cofactor expansions. 2. **Determinant of the Product of Matrices:** Using the property of determinants that for any scalar \( k \) and matrices \( M \): \[ \text{det}(kA) = k^n \text{det}(A) \] where \( n \) is the order of the square matrix \( A \). And using the fact that \( \text{det}(AB) = \
### Matrix Determinants and Operations **Problem Statement:** Suppose that the matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) has determinant \( \text{det}(A) = -3 \). And the matrix \( B = \begin{bmatrix} x & b & c \\ y & e & f \\ z & h & i \end{bmatrix} \) has determinant \( \text{det}(B) = 2 \). **Objective:** 1. Find the determinant of the following matrix: \[ \text{det} \left( \begin{bmatrix} a + 5x & c & b \\ d + 5y & f & e \\ g + 5z & j & h \end{bmatrix} \right) \] 2. Find the determinant of \( -2A^2B^{-1} \). **Details for Explanation:** 1. The matrix inside the determinant in the first problem has elements adjusted by adding 5 times the corresponding elements of another matrix. 2. For the second problem, the operation involves finding \( A^2 \) (the square of matrix \( A \)), multiplying it by \(-2\), and then multiplying by the inverse of matrix \( B \) (denoted \( B^{-1} \)). --- ### Solution Approach: 1. **Determinant of the Sum Adjusted Matrix:** The given matrix is: \[ \begin{bmatrix} a + 5x & c & b \\ d + 5y & f & e \\ g + 5z & j & h \end{bmatrix} \] To solve this, we need to apply properties of determinants, row operations and possibly cofactor expansions. 2. **Determinant of the Product of Matrices:** Using the property of determinants that for any scalar \( k \) and matrices \( M \): \[ \text{det}(kA) = k^n \text{det}(A) \] where \( n \) is the order of the square matrix \( A \). And using the fact that \( \text{det}(AB) = \
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Determinants
Section3.1: The Determinants Of A Matrix
Problem 70E: The determinant of a 22 matrix involves two products. The determinant of a 33 matrix involves six...
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Question
![### Matrix Determinants and Operations
**Problem Statement:**
Suppose that the matrix \( A = \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix} \) has determinant \( \text{det}(A) = -3 \).
And the matrix \( B = \begin{bmatrix}
x & b & c \\
y & e & f \\
z & h & i
\end{bmatrix} \) has determinant \( \text{det}(B) = 2 \).
**Objective:**
1. Find the determinant of the following matrix:
\[
\text{det} \left( \begin{bmatrix}
a + 5x & c & b \\
d + 5y & f & e \\
g + 5z & j & h
\end{bmatrix} \right)
\]
2. Find the determinant of \( -2A^2B^{-1} \).
**Details for Explanation:**
1. The matrix inside the determinant in the first problem has elements adjusted by adding 5 times the corresponding elements of another matrix.
2. For the second problem, the operation involves finding \( A^2 \) (the square of matrix \( A \)), multiplying it by \(-2\), and then multiplying by the inverse of matrix \( B \) (denoted \( B^{-1} \)).
---
### Solution Approach:
1. **Determinant of the Sum Adjusted Matrix:**
The given matrix is:
\[
\begin{bmatrix}
a + 5x & c & b \\
d + 5y & f & e \\
g + 5z & j & h
\end{bmatrix}
\]
To solve this, we need to apply properties of determinants, row operations and possibly cofactor expansions.
2. **Determinant of the Product of Matrices:**
Using the property of determinants that for any scalar \( k \) and matrices \( M \):
\[
\text{det}(kA) = k^n \text{det}(A)
\]
where \( n \) is the order of the square matrix \( A \).
And using the fact that \( \text{det}(AB) = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ed4e6f2-ec2f-4b59-835f-0c83565bb723%2F21564d05-f3ad-427d-b1fb-3661cda5c71e%2Fnw1gfia_processed.png&w=3840&q=75)
Transcribed Image Text:### Matrix Determinants and Operations
**Problem Statement:**
Suppose that the matrix \( A = \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix} \) has determinant \( \text{det}(A) = -3 \).
And the matrix \( B = \begin{bmatrix}
x & b & c \\
y & e & f \\
z & h & i
\end{bmatrix} \) has determinant \( \text{det}(B) = 2 \).
**Objective:**
1. Find the determinant of the following matrix:
\[
\text{det} \left( \begin{bmatrix}
a + 5x & c & b \\
d + 5y & f & e \\
g + 5z & j & h
\end{bmatrix} \right)
\]
2. Find the determinant of \( -2A^2B^{-1} \).
**Details for Explanation:**
1. The matrix inside the determinant in the first problem has elements adjusted by adding 5 times the corresponding elements of another matrix.
2. For the second problem, the operation involves finding \( A^2 \) (the square of matrix \( A \)), multiplying it by \(-2\), and then multiplying by the inverse of matrix \( B \) (denoted \( B^{-1} \)).
---
### Solution Approach:
1. **Determinant of the Sum Adjusted Matrix:**
The given matrix is:
\[
\begin{bmatrix}
a + 5x & c & b \\
d + 5y & f & e \\
g + 5z & j & h
\end{bmatrix}
\]
To solve this, we need to apply properties of determinants, row operations and possibly cofactor expansions.
2. **Determinant of the Product of Matrices:**
Using the property of determinants that for any scalar \( k \) and matrices \( M \):
\[
\text{det}(kA) = k^n \text{det}(A)
\]
where \( n \) is the order of the square matrix \( A \).
And using the fact that \( \text{det}(AB) = \
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