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May 9th, 2015, 11:12 AM   #1
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Relationship between the rows of three matrices

Hello, I'm having trouble with the following problem:

We solve a system of homogenous linear equations given by the matrix A and we write the basis of its solution space into the rows of the matrix B.
Next, we solve the system Bx=0 and we write the basis of its solution space into the rows of matrix C.

What is the relationship in between the rows of matrices A, B and C?

1) The rows of C are linear combinations of the rows of A, but it is possible that a row of A exists such that it is not a linear combination of the rows of C.

2)The rows of B are linear combinations of the rows of A or the rows of C.

3)The linear span of the rows of A is equivalent to that of rows of C.

4)The number of rows of C is greater than the number of rows of A.

5) Nothing above is generally true.

I hold a strong suspicion that the answer is 5). The reasoning behind this is the fact that the matrix A might be regular, and therefore the dimension of its solution space is 0 => we don't write anything into B nor C and that invalidates every answer apart from 5). However, I am not entirely sure, so I would like to get a confirmation or refutation.

Any input would be greatly welcome, thanks!
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