# 2 non-zero matrix A and B, and A*B=0

• blaster
In summary, when two non-zero matrices, A and B, have a product of 0, it means that the resulting matrix from multiplying A and B is a matrix of all zeros. There are infinite possible values for A and B if A*B=0, as long as the elements of A and B are chosen in a way that results in a matrix of all zeros. A and B can be square matrices even if A*B=0, and it is possible for either A or B to be the zero matrix in this scenario. The significance of A and B being non-zero matrices is that it can indicate a relationship between the two matrices, which can be useful in solving systems of equations or understanding the behavior of a system.
blaster
There are 2 non-zero matrix A and B, and A*B=0.
another matrix C is a row equivalent to A.
is C*B=0 ?

Yes. Because C=E*A for some matrix E.

actually this is the fundamental reasaon row equivalence is a useful notion in solving linear systems.

i.e. this says exactly that any solutions of the equation AX=0, also solve CX=0.

recall the use of row reduction in solving systems: you reduce A to an equivalent matrix such that CX=0 is easy to solve. then you announce that the solutions are also solutions of AX=0.

## 1. What does it mean when two non-zero matrices, A and B, have a product of 0?

When two non-zero matrices, A and B, have a product of 0, it means that the resulting matrix from multiplying A and B is a matrix of all zeros. This does not necessarily mean that A or B individually are equal to 0.

## 2. What are some possible values for A and B if A*B=0?

There are infinite possible values for A and B if A*B=0. As long as the elements of A and B are chosen in a way that results in a matrix of all zeros when multiplied together, any values can be used.

## 3. Can A and B be square matrices if A*B=0?

Yes, A and B can be square matrices if A*B=0. The dimensions of A and B do not need to match for their product to equal 0.

## 4. Is it possible for A or B to be the zero matrix if A*B=0?

Yes, it is possible for either A or B to be the zero matrix if A*B=0. However, it is not necessary for either matrix to be zero for their product to equal 0.

## 5. What is the significance of A and B being non-zero matrices in this scenario?

If A and B are both non-zero matrices and their product is 0, it can indicate that there is a dependence or relationship between the two matrices. This could be useful in solving systems of equations or understanding the behavior of a system.

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