Matrices concept confusion....

Click For Summary
SUMMARY

The discussion centers around the confusion regarding the matrix equation B^3 + B = 2B^2 + 2I. The user incorrectly concludes that B must equal 2I, failing to account for the possibility that the matrix (B^2 + I) may be singular and thus non-invertible. The correct interpretation emphasizes the importance of matrix invertibility in solving equations involving matrices. The conclusion drawn is that the existence of the inverse of (B^2 + I) is crucial to validating the user's assumption.

PREREQUISITES
  • Understanding of matrix algebra, specifically matrix multiplication and addition.
  • Knowledge of matrix inverses and conditions for invertibility.
  • Familiarity with singular matrices and their properties.
  • Basic concepts of linear transformations and their representation in matrix form.
NEXT STEPS
  • Study the properties of singular matrices and conditions for matrix invertibility.
  • Learn about the implications of matrix equations in linear algebra.
  • Explore the concept of matrix transformations and their geometric interpretations.
  • Review examples of matrix equations and their solutions to solidify understanding.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to clarify concepts related to matrix equations and invertibility.

Clara Chung
Messages
300
Reaction score
13
In the photo that is a matrix B. B^3 + B = 2B^2 +2I from calculations. But why not B must be equal to 2I, the reason is as follow,
B(B^2 + I ) = 2(B^2 +I)
B(B^2 + I)(B^2 + I)^(-1) = 2I (B^2 + I)(B^2 + I)^(-1)
so B = 2I why is my concept wrong? please explain to me.
 

Attachments

  • Untitled.png
    Untitled.png
    5 KB · Views: 463
Physics news on Phys.org
Your concept assumes that the matrix (B^2 + I)^-1 exists. But there is also the possibility that the matrix (B^2 + I) is singular and has no inverse, which is the case for your matrix B.
 
  • Like
Likes   Reactions: mfb and Clara Chung

Similar threads

Undergrad Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Why are determinants in 2x2 matrices and 3x3 matrices computed the way they are?
  • · Replies 13 ·
Replies
13
Views
4K
High School Proving the uniqueness of eigenspaces
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad Solving a complex linear system with parameters
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad The vector to which a dual vector corresponds
  • · Replies 13 ·
Replies
13
Views
5K
Undergrad Applying change of basis to vector b_1 doesn't give b'_1?
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad What is the proper matrix product?
  • · Replies 9 ·
Replies
9
Views
3K
MHB Solving for Invertible Matrix: What Am I Doing Wrong?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Markov Process Limit: Calculating $u_k$ as a Function of $a,b$
  • · Replies 1 ·
Replies
1
Views
2K
MHB What Are the Minimal Polynomials of Matrix Powers?
  • · Replies 4 ·
Replies
4
Views
2K