Can you always find B such that AB=0 when A is Noninvertible?

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In summary, the conversation discusses the possibility of finding a nonzero matrix B that can be multiplied with a noninvertible matrix A to result in a zero matrix. It is suggested that this can be achieved by taking advantage of the fact that a noninvertible matrix has at least one column or row with all zeros. Thus, by setting the corresponding row or column in matrix B to be non-zero and the rest to be zeros, the product of A and B will always be zero. It is mentioned that this approach may not work if A is not row reduced, but it provides a possible solution to the problem.
  • #1
buzzmath
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Could anyone help me with this problem?

If A is a noninvertible nxn matrix, can you always find a nonzero nxn matrix B such that AB = 0?

I think you will always be able to find this because a noninvertible matrix can't be reduced to the identity matrix so when row reduced it either has a column or a row that is all zeros. if it's a column that's all zeros say column i then row i in matrix B can be whatever you want and everything else can be zeros. and if row i is all zeros then there will always be a free variable in the matrix to make the product of the two matrices zero. I'm not sure if these are the only cases, if this could really work if the matrix A isn't row reduced, or if this is even a proof or close. Thanks
 
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  • #2
Don't worry about row reducing anything. What can you say about the columns of A? Go from there.
 

1. What does it mean for a matrix to be noninvertible?

A matrix is noninvertible if it does not have an inverse matrix. This means that there is no matrix that can be multiplied with the original matrix to produce the identity matrix (a square matrix with 1s on the main diagonal and 0s elsewhere).

2. Why is it important to know if a matrix is noninvertible?

Knowing if a matrix is noninvertible is important because it determines if the matrix has a unique solution or if there are multiple solutions or no solutions at all. It also affects the efficiency of solving linear equations using matrix operations.

3. Can you always find a B matrix such that AB=0 when A is noninvertible?

No, it is not always possible to find a B matrix such that AB=0 when A is noninvertible. This is because the only matrix that can satisfy this equation is the zero matrix, which is not always a valid solution.

4. Is there a way to determine if a noninvertible matrix has a solution for AB=0?

Yes, a noninvertible matrix will have a solution for AB=0 if and only if the null space (also known as the kernel) of the matrix A is nonempty. This means that there is at least one nontrivial solution that satisfies the equation.

5. How can I find a B matrix that satisfies AB=0 for a noninvertible matrix A?

To find a B matrix that satisfies AB=0 for a noninvertible matrix A, you can use the null space of A. The null space contains all the vectors that when multiplied by A, will result in the zero vector. These vectors can be used as the columns of the B matrix to satisfy the equation.

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