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1. Let A, B € Rnxn and assuming both A-¹ B-¹ exist. Show that (AB)-¹ = B-¹A-¹. 2. Let A € Rnxn. Show that if Ax 0 for all x ER", then A = 0. 3. Let A € Rnxn. Show that if A is strictly lower triangular (i.e. aj = 0 if i ≤j) then A" = 0. (use induction). 4. Let A € Rnxn, u, v E Rn. Compute the inverse of A = I + uvT. Hint: try a matrix I - auv with a R to be determined. What constraint must u and v satisfy for the inverse to exist?

1. Let A, B € Rnxn and assuming both A-¹ B-¹ exist. Show that (AB)-¹ = B-¹A-¹. 2. Let A € Rnxn. Show that if Ax 0 for all x ER", then A = 0. 3. Let A € Rnxn. Show that if A is strictly lower triangular (i.e. aj = 0 if i ≤j) then A" = 0. (use induction). 4. Let A € Rnxn, u, v E Rn. Compute the inverse of A = I + uvT. Hint: try a matrix I - auv with a R to be determined. What constraint must u and v satisfy for the inverse to exist?

Algebra and Trigonometry (6th Edition)
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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1. Let \( A, B \in \mathbb{R}^{n \times n} \) and assuming both \( A^{-1} \) and \( B^{-1} \) exist. Show that \( (AB)^{-1} = B^{-1} A^{-1} \). 2. Let \( A \in \mathbb{R}^{n \times n} \). Show that if \( Ax = 0 \) for all \( x \in \mathbb{R}^n \), then \( A = 0 \). 3. Let \( A \in \mathbb{R}^{n \times n} \). Show that if \( A \) is strictly lower triangular (i.e. \( a_{ij} = 0 \) if \( i \leq j \)) then \( A^n = 0 \). (use induction). 4. Let \( A \in \mathbb{R}^{n \times n}, u, v \in \mathbb{R}^n \). Compute the inverse of \( A = I + uv^T \). Hint: try a matrix \( I - \alpha uv^T \) with \( \alpha \in \mathbb{R} \) to be determined. What constraint must \( u \) and \( v \) satisfy for the inverse to exist?
Transcribed Image Text:1. Let \( A, B \in \mathbb{R}^{n \times n} \) and assuming both \( A^{-1} \) and \( B^{-1} \) exist. Show that \( (AB)^{-1} = B^{-1} A^{-1} \). 2. Let \( A \in \mathbb{R}^{n \times n} \). Show that if \( Ax = 0 \) for all \( x \in \mathbb{R}^n \), then \( A = 0 \). 3. Let \( A \in \mathbb{R}^{n \times n} \). Show that if \( A \) is strictly lower triangular (i.e. \( a_{ij} = 0 \) if \( i \leq j \)) then \( A^n = 0 \). (use induction). 4. Let \( A \in \mathbb{R}^{n \times n}, u, v \in \mathbb{R}^n \). Compute the inverse of \( A = I + uv^T \). Hint: try a matrix \( I - \alpha uv^T \) with \( \alpha \in \mathbb{R} \) to be determined. What constraint must \( u \) and \( v \) satisfy for the inverse to exist?
5. **Using the results obtained in 1) and 4)** and assuming \( A \in \mathbb{R}^{n \times n} \), \( u, v \in \mathbb{R}^n \), and \( A^{-1} \) exists, show that \[ (A + uv^T)^{-1} = A^{-1} - \frac{1}{1 + v^T A^{-1} u} A^{-1} uv^T A^{-1} \]
Transcribed Image Text:5. **Using the results obtained in 1) and 4)** and assuming \( A \in \mathbb{R}^{n \times n} \), \( u, v \in \mathbb{R}^n \), and \( A^{-1} \) exists, show that \[ (A + uv^T)^{-1} = A^{-1} - \frac{1}{1 + v^T A^{-1} u} A^{-1} uv^T A^{-1} \]
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