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Consider an 5 x 5 matrix A satisfying the following two properties: (a) Every row has exactly one non-zero entry and it's equal to 1. (b) Every column has exactly one non-zero entry and it's equal to 1. Show that such an A must be invertible.

Consider an 5 x 5 matrix A satisfying the following two properties: (a) Every row has exactly one non-zero entry and it's equal to 1. (b) Every column has exactly one non-zero entry and it's equal to 1. Show that such an A must be invertible.

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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Consider an 5 x 5 matrix A satisfying the following two properties: (a) Every row has exactly one non-zero entry and it's equal to 1. (b) Every column has exactly one non-zero entry and it's equal to 1. Show that such an A must be invertible. Hint: There are 5! = 120 such matrices (Why?). Don't do a case by case analysis. You can try first 2 x 2 and 3 x 3 matrices to get grip on the problem (there are only 2 matrices in the first case and 6 in the second).
Transcribed Image Text:Consider an 5 x 5 matrix A satisfying the following two properties: (a) Every row has exactly one non-zero entry and it's equal to 1. (b) Every column has exactly one non-zero entry and it's equal to 1. Show that such an A must be invertible. Hint: There are 5! = 120 such matrices (Why?). Don't do a case by case analysis. You can try first 2 x 2 and 3 x 3 matrices to get grip on the problem (there are only 2 matrices in the first case and 6 in the second).
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