Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
1.4 19 and 22 please on paper ( theyr very short thank youu)
![---
### Chapter 1: Systems of Linear Equations and Matrices
---
#### Exercises 21-28
**21.**
\[ A = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix} \]
**22.**
\[ A = \begin{bmatrix} 2 & 0 \\ 4 & 1 \end{bmatrix} \]
In Exercises 23-24, let
\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} \]
**23.** Find all values of \( a \), \( b \), \( c \), and \( d \) (if any) for which the matrices \( A \) and \( B \) commute.
**24.** Find all values of \( a \), \( b \), \( c \), and \( d \) (if any) for which the matrices \( A \) and \( C \) commute.
In Exercises 25-28, use the method of Example 8 to find the unique solution of the given linear system.
**25.**
\[ \begin{aligned}
3x_{1} - 2x_{2} &= -1 \\
4x_{1} + 5x_{2} &= 3
\end{aligned} \]
**26.**
\[ \begin{aligned}
-x_{1} + 5x_{2} &= 4 \\
-x_{1} - 3x_{2} &= 1
\end{aligned} \]
**28.**
\[ \begin{aligned}
2x_{1} - 2x_{2} &= 4 \\
x_{1} + 4x_{2} &= 4
\end{aligned} \]
A polynomial \( p(x) \) can be factored as a product of lower degree polynomials, say
\[ p(x) = p_1(x)p_2(x) \]
If \( A \) is a square matrix, then it can be proved that
\[ p(A) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48922dd6-b6f8-4fac-84c6-05a7db5750f6%2F06da5c67-2364-44a4-a1aa-3af739c4cf5c%2Freg044p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Chapter 1: Systems of Linear Equations and Matrices
---
#### Exercises 21-28
**21.**
\[ A = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix} \]
**22.**
\[ A = \begin{bmatrix} 2 & 0 \\ 4 & 1 \end{bmatrix} \]
In Exercises 23-24, let
\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} \]
**23.** Find all values of \( a \), \( b \), \( c \), and \( d \) (if any) for which the matrices \( A \) and \( B \) commute.
**24.** Find all values of \( a \), \( b \), \( c \), and \( d \) (if any) for which the matrices \( A \) and \( C \) commute.
In Exercises 25-28, use the method of Example 8 to find the unique solution of the given linear system.
**25.**
\[ \begin{aligned}
3x_{1} - 2x_{2} &= -1 \\
4x_{1} + 5x_{2} &= 3
\end{aligned} \]
**26.**
\[ \begin{aligned}
-x_{1} + 5x_{2} &= 4 \\
-x_{1} - 3x_{2} &= 1
\end{aligned} \]
**28.**
\[ \begin{aligned}
2x_{1} - 2x_{2} &= 4 \\
x_{1} + 4x_{2} &= 4
\end{aligned} \]
A polynomial \( p(x) \) can be factored as a product of lower degree polynomials, say
\[ p(x) = p_1(x)p_2(x) \]
If \( A \) is a square matrix, then it can be proved that
\[ p(A) =
![# Educational Website Transcription: Matrix and Polynomial Operations
## Problem 10
Find the inverse of the matrix:
\[ \begin{bmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{bmatrix} \]
## Exercises 11-14
Verify that the equations are valid for the matrices in Exercises 5-8.
**Exercise 11:**
\[ (A^T)^{-1} = (A^{-1})^T \]
**Exercise 12:**
\[ (A^{-1})^{-1} = A \]
**Exercise 13:**
\[ (ABC)^{-1} = C^{-1}B^{-1}A^{-1} \]
**Exercise 14:**
\[ (ABC)^T = C^TB^TA^T \]
## Exercises 15-18
Use the given information to find matrix \( A \).
**Exercise 15:**
\[ (7A)^{-1} = \begin{bmatrix}
-3 & 7 \\
1 & -2
\end{bmatrix} \]
**Exercise 16:**
\[ (5A^T)^{-1} = \begin{bmatrix}
-3 & -1 \\
5 & -2
\end{bmatrix} \]
**Exercise 17:**
\[ (I + 2A)^{-1} = \begin{bmatrix}
1 & 2 \\
4 & 5
\end{bmatrix} \]
**Exercise 18:**
\[ A^{-1} = \begin{bmatrix}
2 & -1 \\
3 & -5
\end{bmatrix} \]
## Exercises 19-20
Compute the following using the given matrix \( A \).
**Exercise 19:**
\[ A = \begin{bmatrix}
3 & 1 \\
2 & 1
\end{bmatrix} \]
- **Part a:** \( A^3 \)
- **Part b:** \( A^{-3} \)
- **Part c:** \( A^2 - 2A + I \)
**Exercise 20:**
\[ A = \begin{bmatrix}
2 & 0 \\
4 & 1
\end{bmatrix} \]
## Exercises 21-22
Compute \( p(A) \) for the given](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48922dd6-b6f8-4fac-84c6-05a7db5750f6%2F06da5c67-2364-44a4-a1aa-3af739c4cf5c%2Fe5oiuyw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Educational Website Transcription: Matrix and Polynomial Operations
## Problem 10
Find the inverse of the matrix:
\[ \begin{bmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{bmatrix} \]
## Exercises 11-14
Verify that the equations are valid for the matrices in Exercises 5-8.
**Exercise 11:**
\[ (A^T)^{-1} = (A^{-1})^T \]
**Exercise 12:**
\[ (A^{-1})^{-1} = A \]
**Exercise 13:**
\[ (ABC)^{-1} = C^{-1}B^{-1}A^{-1} \]
**Exercise 14:**
\[ (ABC)^T = C^TB^TA^T \]
## Exercises 15-18
Use the given information to find matrix \( A \).
**Exercise 15:**
\[ (7A)^{-1} = \begin{bmatrix}
-3 & 7 \\
1 & -2
\end{bmatrix} \]
**Exercise 16:**
\[ (5A^T)^{-1} = \begin{bmatrix}
-3 & -1 \\
5 & -2
\end{bmatrix} \]
**Exercise 17:**
\[ (I + 2A)^{-1} = \begin{bmatrix}
1 & 2 \\
4 & 5
\end{bmatrix} \]
**Exercise 18:**
\[ A^{-1} = \begin{bmatrix}
2 & -1 \\
3 & -5
\end{bmatrix} \]
## Exercises 19-20
Compute the following using the given matrix \( A \).
**Exercise 19:**
\[ A = \begin{bmatrix}
3 & 1 \\
2 & 1
\end{bmatrix} \]
- **Part a:** \( A^3 \)
- **Part b:** \( A^{-3} \)
- **Part c:** \( A^2 - 2A + I \)
**Exercise 20:**
\[ A = \begin{bmatrix}
2 & 0 \\
4 & 1
\end{bmatrix} \]
## Exercises 21-22
Compute \( p(A) \) for the given
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