Linear algebra: please solve q27 and 28 correctly and handwritten. Theoram 2 is also attached THEOREM 2EXAMPLE 8Determinants and Singular MatricesTheorems 2 and 3, which follow, are fundamental to our study of eigenvalues. Thesetheorems are stated here and their proofs are given in Chapter 6.Let A and B be (n x n) matrices. Thendet (AB) = det (A) det(B).The following example illustrates Theorem 2.Calculate det (A), det (B), and det(AB) for the matrices31 25. Suppose that A is an (n × n) nonsingular matrix,and recall that det (I) = 1, where I is the (n x n)identity matrix. Show that det(A-¹) = 1/det(A).26. If A and B are (n × n) matrices, then usually AB +BA. Nonetheless, argue that always det (AB)det (BA).=In Exercises 27-30, use Theorem 2 and Exercise 25to evaluate the given determinant, where A and B are(n = n) matrices with det(A) = 3 and det(B) = 5.27. det(ABA-¹)28. det(A²B)29. det(A-¹B-¹A²)30. det(AB-¹A-¹B)

Given, and.We solve these questions using theorem 2 and Exercise 25.

Answered: In Exercises 27-30, use Theorem 2 and…

In Exercises 27-30, use Theorem 2 and Exercise 25 to evaluate the given determinant, where A and B are (n x n) matrices with det(A) = 3 and det (B) = 5. 27. det(ABA-¹) 28. det(A²B)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 10AEXP

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Linear algebra: please solve q27 and 28 correctly and handwritten. Theoram 2 is also attached

THEOREM 2
EXAMPLE 8
Determinants and Singular Matrices
Theorems 2 and 3, which follow, are fundamental to our study of eigenvalues. These
theorems are stated here and their proofs are given in Chapter 6.
Let A and B be (n x n) matrices. Then
det (AB) = det (A) det(B).
The following example illustrates Theorem 2.
Calculate det (A), det (B), and det(AB) for the matrices
31

25. Suppose that A is an (n × n) nonsingular matrix,
and recall that det (I) = 1, where I is the (n x n)
identity matrix. Show that det(A-¹) = 1/det(A).
26. If A and B are (n × n) matrices, then usually AB +
BA. Nonetheless, argue that always det (AB)
det (BA).
=
In Exercises 27-30, use Theorem 2 and Exercise 25
to evaluate the given determinant, where A and B are
(n = n) matrices with det(A) = 3 and det(B) = 5.
27. det(ABA-¹)
28. det(A²B)
29. det(A-¹B-¹A²)
30. det(AB-¹A-¹B)

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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