a. Explanation Given: Let A be anidempotentmatrix. Show that I − A is also idempotent. Calculation: A matrix A is said to be idempotent if A 2 = A . Let A be an idempotent matrix and A = [ 1 0 1 0 ] Show that I − A is also idempotent. In order to show that I − A is also idempotent firsttake an identity matrix I of the same order that of matrix A . As matrix A is a 2 × 2 matrix, take a 2 × 2 Identity matrix I = [ 1 0 0 1 ] Now find I − A . In order to subtract two matrices, subtract the corresponding entries. I − A = [ 1 0 0 1 ] − [ 1 0 1 0 ] I − A = [ 1 − 1 0 − 0 0 − 1 1 − 0 ] ( S u b t r a c t t w o m a t r i c e s ) I − A = [ 0 0 − 1 1 ] I − A = [ 0 0 − 1 1 ] I − A is idempotent if ( I − A ) 2 = I − A . Now find ( I − A ) 2 ( I − A ) 2 = ( I − A ) × ( I − A ) ( I − A ) 2 = [ 0 0 − 1 1 ] × [ 0 0 − 1 1 ] In order to multiply two matrices, take the first row of first matrix and multiply it with first Column of second matrix b. To determine I + A is nonsingular and ( I + A ) − 1 = I − 1 2 A

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

9th Edition

ISBN: 9780321962218

Author: Steven J. Leon

Publisher: PEARSON

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Question

Chapter 1.4, Problem 25E

To determine

I−A is idempotent.

To determine

I+A is nonsingular and (I+A)−1=I−12A

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Chapter 1.4, Problem 24E

Chapter 1.4, Problem 26E

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Chapter 1 Solutions

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

Ch. 1.1 - Use back substitution to solve each of the...Ch. 1.1 - Write out the coefficient matrix for each of the...Ch. 1.1 - In each of the following systems, interpret each...Ch. 1.1 - Write an augmented matrix for each of the systems...Ch. 1.1 - Write out the system of equations that corresponds...Ch. 1.1 - Solve each of the following systems. (a)...Ch. 1.1 - The two systems 2x1+x2=34x1+3x2=5 and...Ch. 1.1 - Solve the two systems...Ch. 1.1 - Given a System of the form m1x1+x2=b1m2x1+x2=b2...Ch. 1.1 - Consider a system of the form...

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I − A is idempotent.

Solution Summary: The author proves that I-A is idempotent. Let A be an ilmtmatrix. Then take a 2times 2 identity matrix.

I−A is idempotent.

I+A is nonsingular and (I+A)−1=I−12A

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Chapter 1 Solutions