46. A square matrix A is said to be idempotent if A² = A. a. Show that if A is idempotent, then so is I - A. b. Show that if A is idempotent, then 2A - I is invertible and is its own inverse.
46. A square matrix A is said to be idempotent if A² = A. a. Show that if A is idempotent, then so is I - A. b. Show that if A is idempotent, then 2A - I is invertible and is its own inverse.
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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F1.4 question 46 on paper please
![### Advanced Matrix Theory Problems
Welcome to our educational portal on advanced matrix theory. Below, you'll find a series of problems designed to deepen your understanding of matrix algebra, invertibility, and specific matrix properties. Follow along to challenge your knowledge and enhance your mathematical skills.
#### Problem Set
**43.**
**a.** Show that if \( A \) is invertible and \( AB = AC \), then \( B = C \).
**b.** Explain why part (a) and Example 3 do not contradict one another.
**44.** Show that if \( A \) is invertible and \( k \) is any nonzero scalar, then \( (kA)^n = k^n A^n \) for all integer values of \( n \).
**45.**
**a.** Show that if \( A, B,\) and \( A + B \) are invertible matrices with the same size, then
\[ A(A^{-1} + B^{-1})B(A + B)^{-1} = I. \]
**b.** What does the result in part (a) tell you about the matrix \( A^{-1} + B^{-1} \)?
**46.** A square matrix \( A \) is said to be **idempotent** if \( A^2 = A \).
**a.** Show that if \( A \) is idempotent, then so is \( I - A \).
**b.** Show that if \( A \) is idempotent, then \( 2A - I \) is invertible and its own inverse.
**47.** Show that if \( A \) is a square matrix such that \( A^k = 0 \) for some positive integer \( k \), then the matrix \( I - A \) is invertible and
\[ (I - A)^{-1} = I + A + A^2 + \cdots + A^{k-1}. \]
**48.** Show that the matrix
\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
satisfies the equation
\[ A^2 - (a + d)A + (ad - bc)I = 0. \]
**49.** Assuming that all matrices are \( n \times n \) and invertible, solve for \( D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccdd90b5-cd41-4340-8a36-15b102c9ba73%2Fe2b0bf22-108a-481f-8c98-422cb79db810%2Fxa84d33_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Advanced Matrix Theory Problems
Welcome to our educational portal on advanced matrix theory. Below, you'll find a series of problems designed to deepen your understanding of matrix algebra, invertibility, and specific matrix properties. Follow along to challenge your knowledge and enhance your mathematical skills.
#### Problem Set
**43.**
**a.** Show that if \( A \) is invertible and \( AB = AC \), then \( B = C \).
**b.** Explain why part (a) and Example 3 do not contradict one another.
**44.** Show that if \( A \) is invertible and \( k \) is any nonzero scalar, then \( (kA)^n = k^n A^n \) for all integer values of \( n \).
**45.**
**a.** Show that if \( A, B,\) and \( A + B \) are invertible matrices with the same size, then
\[ A(A^{-1} + B^{-1})B(A + B)^{-1} = I. \]
**b.** What does the result in part (a) tell you about the matrix \( A^{-1} + B^{-1} \)?
**46.** A square matrix \( A \) is said to be **idempotent** if \( A^2 = A \).
**a.** Show that if \( A \) is idempotent, then so is \( I - A \).
**b.** Show that if \( A \) is idempotent, then \( 2A - I \) is invertible and its own inverse.
**47.** Show that if \( A \) is a square matrix such that \( A^k = 0 \) for some positive integer \( k \), then the matrix \( I - A \) is invertible and
\[ (I - A)^{-1} = I + A + A^2 + \cdots + A^{k-1}. \]
**48.** Show that the matrix
\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
satisfies the equation
\[ A^2 - (a + d)A + (ad - bc)I = 0. \]
**49.** Assuming that all matrices are \( n \times n \) and invertible, solve for \( D
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