(a) Explanation Given information: The matrixis A = [ A 11 A 12 0 A 22 ] . Concept used: A matrix is a collection of numbers arranged into a fixed number of rows and columns. Usually the numbers are real numbers. Let A be a singular matrix then there exist a non- vector B such that A B = 0 . Consider the given matrix. A = [ A 11 A 12 0 A 22 ] Let A 11 and A 22 are n × n non-singular matrices. If A is non-singular if det A ≠ 0 . det A = det ( [ A 11 A 12 0 A 22 ] ) = det ( A 11 A 22 − A 12 ⋅ 0 ) = det ( A 11 A 22 ) = det ( A 11 ) det ( A 22 ) ... ( 1 ) As A 11 and A 22 are n × n non-singular matrices. Thus, det ( A 11 ) ≠ 0 and det ( A 22 ) ≠ 0 . This implies, from ( 1 ) , det ( A ) ≠ 0 . Hence, A is non-singular. As A 11 and A 22 are n × n non-singular matrices, therefore, there exist a non-singular matrices A 11 − 1 and A 22 − 1 such that A 11 A 11 − 1 = A 11 − 1 A 11 = I n × n and A 22 A 22 − 1 = A 22 − 1 A 22 = I n × n . Assume that B = [ P Q 0 S ] is the inverse of A such that A B = B A = I (b) To determine To find: the value of C .

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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Chapter 1.6, Problem 11E

(a)

To determine

To Show: A must also be non-singular and that A−1 must be of the form [ A 11 −1CO A 22 −1] .

(b)

To determine

To find: the value of C .

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Chapter 1.6, Problem 10E

Chapter 1.6, Problem 12E

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Solve the linear system of equations attached using Gaussian elimination (not Gauss-Jordan) and back subsitution. Remember that: A matrix is in row echelon form if Any row that consists only of zeros is at the bottom of the matrix. The first non-zero entry in each other row is 1. This entry is called aleading 1. The leading 1 of each row, after the first row, lies to the right of the leading 1 of the previous row.

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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A must also be non-singular and that A − 1 must be of the form [ A 11 − 1 C O A 22 − 1 ] .

Solution Summary: The author explains that the matrix is a collection of numbers arranged into fixed number of rows and columns.

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To Show: A must also be non-singular and that A−1 must be of the form [ A 11 −1CO A 22 −1] .

To find: the value of C .

Want to see the full answer?

Chapter 1 Solutions