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Let A and B be n x n matrices. 1. If A and B are anti-symmetric matrices that commute under multiplication, show that their product AB is symmetric. 2. A matrix A is called nilpotent of degree 2 if A² = 0. Show that if A is nilpotent of degree 2, then the matrix (I - A) is invertible, and find an expression for its inverse.

Let A and B be n x n matrices. 1. If A and B are anti-symmetric matrices that commute under multiplication, show that their product AB is symmetric. 2. A matrix A is called nilpotent of degree 2 if A² = 0. Show that if A is nilpotent of degree 2, then the matrix (I - A) is invertible, and find an expression for its inverse.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.1: Operations With Matrices
Problem 76E
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Let A and B be n x n matrices. 1. If A and B are anti-symmetric matrices that commute under multiplication, show that their product AB is symmetric. 2. A matrix A is called nilpotent of degree 2 if A² = 0. Show that if A is nilpotent of degree 2, then the matrix (I - A) is invertible, and find an expression for its inverse.
Transcribed Image Text:Let A and B be n x n matrices. 1. If A and B are anti-symmetric matrices that commute under multiplication, show that their product AB is symmetric. 2. A matrix A is called nilpotent of degree 2 if A² = 0. Show that if A is nilpotent of degree 2, then the matrix (I - A) is invertible, and find an expression for its inverse.
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