Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 27E from Chapter 6.2 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 27E
Chapter:
Problem:
Let U be a square matrix with orthonormal columns. Explain why U is invertible. (Mention the theorems you use.)
Step-by-Step Solution
Given Information
We are given that U is a square matrix with orthonormal columns. We have to prove why U is an invertible matrix.
Step-1:
If U is an orthonormal matrix, then it is a square matrix and \[{U^T}U = I\] By the invertible matrix theorem if there is a square matrix, A such that $ A D = I$, then the matrix is Invertible.
The matrix U is invertible
We are given that U is a square matrix with orthonormal columns. We have to prove why U is an invertible matrix.
Step-1:
If U is an orthonormal matrix, then it is a square matrix and \[{U^T}U = I\] By the invertible matrix theorem if there is a square matrix, A such that $ A D = I$, then the matrix is Invertible.
The matrix U is invertible