Get Expert Help with Invertible Matrices: Tips and Techniques

  • Thread starter Thread starter FancyChancey
  • Start date Start date
  • Tags Tags
    Matrix
Click For Summary
SUMMARY

This discussion focuses on the properties of invertible matrices, specifically addressing the conditions under which matrices are invertible and the implications of certain matrix equations. The user inquires about the invertibility of the expression (A + 2A - 1) and its determinant, confirming that a matrix is invertible if its determinant is non-zero. Additionally, the discussion explores the implications of the equation B² - 2B + I = 0, leading to the conclusion that B - I must be non-invertible under specific conditions.

PREREQUISITES
  • Understanding of matrix theory, specifically invertible matrices
  • Knowledge of determinants and their role in matrix invertibility
  • Familiarity with polynomial equations involving matrices
  • Basic algebraic manipulation of matrix equations
NEXT STEPS
  • Study the properties of determinants in relation to matrix invertibility
  • Learn about the implications of matrix equations like B² - 2B + I = 0
  • Explore examples of non-invertible matrices and their characteristics
  • Investigate the concept of eigenvalues and their relation to matrix invertibility
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to enhance their understanding of matrix properties and invertibility.

FancyChancey
Messages
4
Reaction score
0
Hi all. I'm having a real tough time with this question. I don't know where to begin and how to go about the question. Can someone point me in the right direction please?

(Problem is on the attachment)
 

Attachments

  • #3.jpg
    #3.jpg
    17 KB · Views: 399
Physics news on Phys.org


haven't we seen this before?

a) If A is invertible, then (A+ 2A-1) is invertible.
When does there exist a matrix, B, such that (A+ 2A-1)B= I?
If I remember correctly, a matrix is invertible if and only if it's determinant is not 0. What is the determinant of A+ 2A-1?

b) If a real matrix B satisfies B2- 2B+ 1= 0, then B-I cannot be invertible.
B2- 2B+I= (B- I)2

c) There is a non-invertible matrix, B, satisfying B2- 2B+ I= 0.
As I said, B2-2B+ I= (B-I)2. Therefore B= ? or ?. Is either of those non-invertible?
 

Similar threads

Prove the identity matrix is unique
  • · Replies 69 ·
3
Replies
69
Views
11K
Transformations to both sides of a matrix equation
  • · Replies 25 ·
Replies
25
Views
3K
Left and right inverses of a non-square matrix
  • · Replies 7 ·
Replies
7
Views
4K
Finding Matrices E & F: A Matrix Challenge
  • · Replies 5 ·
Replies
5
Views
2K
Understanding the Span of Matrices: Can Someone Explain This Question to Me?
  • · Replies 2 ·
Replies
2
Views
2K
Solving Plane Equation 3x + 2y -z = 4
  • · Replies 8 ·
Replies
8
Views
3K
Does there exist a 2x2 non-singular matrix with only one 1d eigenspace?
  • · Replies 2 ·
Replies
2
Views
2K
Matrices: word problem, transition matrix
  • · Replies 4 ·
Replies
4
Views
3K
Determine matrix for reflection followed by rotation
  • · Replies 10 ·
Replies
10
Views
2K
High School Row reduction, Gaussian Elimination on augmented matrix
  • · Replies 6 ·
Replies
6
Views
2K