please send complete handwritten solution for Q 5 Read the following definition and example carefully, and then answer the question below.We say (for n > 2) that an n xn matrix A is grumpy if it has n distinct negative" eigenvalues."i.e. if A is an eigenvalue of A, then A < 0.Here is an example and a couple quick consequences of the definition:The matrixis grumpy, as it has eigenvalues -1 and -3.Allgrumpy matrices are diagonalizable (since they have n distinct eigenvalues.)Similarly, all grumpy matrices are also invertible (since they cannot have 0 as an eigenvalue.)Prove that if A is a 2 x 2 grumpy matrix,' then det(A) is equal to the area of the fundamentalparallelogram of TA (i.e. the image of the unit square under the linear transformation TA).Thm

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We say (for n > 2) that an n x n matrix A is grumpy if it has n distinct negative" eigenvalues. "i.e. if A is an eigenvalue of A, then A < 0. Here is an example and a couple quick consequences of the definition: -1 The matrix -2 is grumpy, as it has eigenvalues -1 and -3. -3 All grumpy matrices are diagonalizable (since they have n distinct eigenvalues.) Similarly, all grumpy matrices are also invertible (since they cannot have 0 as an eigenvalue.) Prove that if A is a 2 x 2 grumpy matrix,' then det(A) is equal to the area of the fundamental parallelogram of TA (i.e. the image of the unit square under the linear transformation TA). Thm Def

We say (for n > 2) that an n x n matrix A is grumpy if it has n distinct negative" eigenvalues. "i.e. if A is an eigenvalue of A, then A < 0. Here is an example and a couple quick consequences of the definition: -1 The matrix -2 is grumpy, as it has eigenvalues -1 and -3. -3 All grumpy matrices are diagonalizable (since they have n distinct eigenvalues.) Similarly, all grumpy matrices are also invertible (since they cannot have 0 as an eigenvalue.) Prove that if A is a 2 x 2 grumpy matrix,' then det(A) is equal to the area of the fundamental parallelogram of TA (i.e. the image of the unit square under the linear transformation TA). Thm Def

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section: Chapter Questions

Problem 1RQ

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