AB=0 but A and B different from 0

  • Thread starter Thread starter bonildo
  • Start date Start date
Click For Summary
To demonstrate that if matrix A is not invertible, there exists a non-zero matrix B such that AB=0, one can utilize the concept of the kernel of A. A non-invertible matrix implies that its columns are linearly dependent, allowing for a nontrivial linear combination of these columns to yield the zero vector. This leads to the construction of matrix B, which can be formed from the linear combinations of the columns of A that result in zero. The hint suggests focusing on the necessary kernel to find this non-zero matrix B. Ultimately, the relationship AB=0 can be established through these linear dependencies.
bonildo
Messages
14
Reaction score
1

Homework Statement


Need to show that if A is not invertible then exist B (nxn) such that AB=0 but B is different from 0.

Homework Equations



The Attempt at a Solution


Thought about writing A like E1...Ek*R where Ei are elementary matrices and R(different from I) the row reduced echelon matrice but doesn't seem.to help, any idea??
 
Physics news on Phys.org
Hint: a nontrivial linear combination of the columns/rows of ##A## is zero.
 
  • Like
Likes Anama Skout
Similar to Micromass, write the columns so that they are in the " necessary ( and non-trivial) " kernel.
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
View full post »

Similar threads

New axis of rotation for composite of rotations in Euclidean space
  • · Replies 4 ·
Replies
4
Views
2K
If A,B are nxn and AB is invertible, then A and B are invertible
  • · Replies 4 ·
Replies
4
Views
5K
Prove that a matrix can be reduced to RRE and CRE
  • · Replies 4 ·
Replies
4
Views
2K
Invertibilility of AB given that B is not invertible
  • · Replies 4 ·
Replies
4
Views
4K
Show that eigenvalue of A + eigvalueof B ≠ eigvalue of A+B?
  • · Replies 28 ·
Replies
28
Views
5K
Show colsp(AB) is contained in colsp(A)
  • · Replies 4 ·
Replies
4
Views
2K
Proving facts about matrices without determinants
  • · Replies 20 ·
Replies
20
Views
3K
Proving that lcm(a,b)gcd(a,b)=ab
  • · Replies 8 ·
Replies
8
Views
4K
The Product of two Unitary Matrices is Unitary Proof
  • · Replies 1 ·
Replies
1
Views
5K
Number of Matrices w/ a+b+c+d=0: Prove 3 Exist
  • · Replies 16 ·
Replies
16
Views
2K