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) = # Educational Website Transcription: Matrix and Polynomial Operations## Problem 10Find the inverse of the matrix:\[ \begin{bmatrix}\cos \theta & \sin \theta \\-\sin \theta & \cos \theta\end{bmatrix} \]## Exercises 11-14Verify 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 15-18Use 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 19-20Compute 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 21-22Compute $ p(A) $ for the given

As you have posted multiple questions, according to the Bartleby guidelines, I will answer first…

Answered: of Linear Equations and 22. A = [²29]

of Linear Equations and 22. A = [²29]

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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