Null matrix and invertible matrix

  • Context: Undergrad 
  • Thread starter Thread starter seema283k
  • Start date Start date
  • Tags Tags
    Matrix
Click For Summary
SUMMARY

The discussion centers on the properties of the null matrix and its relationship with invertible matrices. It clarifies that if A is a null matrix, then for any matrix B, the expression I + AB equals I, which is inherently invertible. Consequently, the theorem stating that I + AB is invertible if and only if I + BA is invertible is confirmed to be vacuously true, as both expressions yield the identity matrix I.

PREREQUISITES
  • Understanding of matrix theory, specifically null matrices.
  • Familiarity with the concept of invertible matrices.
  • Knowledge of matrix operations, including addition and multiplication.
  • Basic comprehension of linear algebra theorems.
NEXT STEPS
  • Study the properties of null matrices in linear algebra.
  • Explore the conditions for matrix invertibility in detail.
  • Learn about the implications of the identity matrix in matrix operations.
  • Investigate other linear algebra theorems related to matrix multiplication.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators looking for clear explanations of matrix properties and theorems.

seema283k
Messages
4
Reaction score
0
how can we use property of null matrix .explain it by giving examples.
please give me explanation of the theorem----if A is a null matrix & B is any matrix then I+AB is invertible iff I+BA is invertible
 
Physics news on Phys.org
Are you sure you mean the null matrix?

Because the null matrix is the matrix with all zero entries.

So if A is the null matrix, then I+AB=I=I+BA. I is invertible. So for your theorem, both the hypothesis and conclusion are true (in both directions). Making the theorem vacuously true.
 

Similar threads

Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
3K
Undergrad Invertible Matrix Proof: A-Transpose * M * A (n by m)
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Spectral theorem for Hermitian matrices-- special cases
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Why Use Gram-Schmidt to Make a Unitary Matrix?
  • · Replies 14 ·
Replies
14
Views
4K
Graduate How do you invert a 3D matrix? (Tensor inversion)
  • · Replies 2 ·
Replies
2
Views
3K
MHB Proving or disproving this matrix V is invertible.
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Matrix rank in terms of determinant polynomial
  • · Replies 14 ·
Replies
14
Views
4K
MHB A and solution are known find B matrix
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form
  • · Replies 7 ·
Replies
7
Views
2K
MHB Solving Systems of Six Equations with Nine Variables
  • · Replies 1 ·
Replies
1
Views
2K