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 1E from Chapter 2.3 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 1E
Chapter:
Problem:
Unless otherwise specified, assume that all matrices in these exercises are n × n. Determine which of the matrices in Exercises 1–10 are invertible. Use as few calculations as possible. Justify your answers.
Step-by-Step Solution
Given Information
We are given with a matrix: \[A = \left[ {\begin{array}{*{20}{c}}5&7\\{ - 3}&{ - 6}\end{array}} \right]\]We have to find whether the matrix is invertible or not.
Step 1: Determinant of the matrix
If the determinant of the matrix is non zero, the matrix is invertible\[\begin{array}{l}\det \left( A \right) = 5 \times \left( { - 6} \right) - 7 \times \left( { - 3} \right)\\\det \left( A \right) = - 30 + 21\\\det \left( A \right) = - 7\end{array}\]Since the determinant of the matrix is nonzero, the matrix is invertibleTherefore,
Matrix is invertible
We are given with a matrix: \[A = \left[ {\begin{array}{*{20}{c}}5&7\\{ - 3}&{ - 6}\end{array}} \right]\]We have to find whether the matrix is invertible or not.
Step 1: Determinant of the matrix
If the determinant of the matrix is non zero, the matrix is invertible\[\begin{array}{l}\det \left( A \right) = 5 \times \left( { - 6} \right) - 7 \times \left( { - 3} \right)\\\det \left( A \right) = - 30 + 21\\\det \left( A \right) = - 7\end{array}\]Since the determinant of the matrix is nonzero, the matrix is invertibleTherefore,
Matrix is invertible