# Proving B=C Given AB=BC and A Non-Singular

• nokia8650
In summary, given the information that AB = BC and A is non-singular, we are asked to prove that B = C. In the second part of the conversation, the task is to find a matrix C with all non-zero elements, but the given matrix A is singular. Therefore, we can construct a C by using the fact that Ax = 0 for some nonzero vector x.
nokia8650
Given that AB = BC, and that A is non-singular, prove that B =C.

I don't know how to do matrices in these forums, so where I write a matrix, I mean (a,b,c,d)

Given that A = (3,6,1,2) and B = (1,5,0,1), find a matrix C whose elements are all non zero.

I can easily do the second part of the question. I cannot understand the logic of the second part of the question - we have proved that B=C, how can C now be non equivalent to B?

Thanks

Because the assumption in the first question is that A is non-singular

nokia8650 said:
Given that AB = BC, and that A is non-singular, prove that B =C.

I don't know how to do matrices in these forums, so where I write a matrix, I mean (a,b,c,d)

Given that A = (3,6,1,2) and B = (1,5,0,1), find a matrix C whose elements are all non zero.

I can easily do the second part of the question. I cannot understand the logic of the second part of the question - we have proved that B=C, how can C now be non equivalent to B?

Thanks

I think you mean AB=AC. Like VeeEight said, in the first problem A is assumed to be nonsingular. The matrix you gave for A is the second part is singular. Ax=0 for some nonzero vector x. Use that to construct a C.

## 1. What does it mean for a matrix to be non-singular?

A non-singular matrix is a square matrix that has a non-zero determinant, meaning it has an inverse. This inverse allows for unique solutions to linear equations.

## 2. How can I prove that B=C if AB=BC and A is non-singular?

You can prove that B=C by using the property of non-singular matrices that states that if AB=AC, then B=C. By setting AB=BC and using the inverse of A, you can simplify the equation to B=C.

## 3. Can I use any non-singular matrix A to prove B=C?

Yes, as long as A is a square matrix with a non-zero determinant, it can be used to prove B=C given AB=BC.

## 4. What is the importance of AB=BC in proving B=C given A is non-singular?

AB=BC is important because it shows that the two matrices are equal up to a scalar multiple. This allows us to use the property of non-singular matrices to prove B=C.

## 5. Are there any other conditions that need to be met in order to prove B=C given AB=BC and A is non-singular?

No, as long as AB=BC and A is non-singular, you can use the property of non-singular matrices to prove B=C. However, it is important to note that this property only applies to square matrices.

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