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I am trying to solve the following problem:

Let A and B be m x n and n x p matrices respectively.

Suppose the homogeneous linear system $ Bx = 0 $ has infinitely many solutions. How many solutions does the system $ ABx = 0 $ have?

From what I know, infinitely many solutions mean that there is at least one non-pivot column in the row-echleon form of a matrix. Let's say we define the following:

$$ A = \begin{bmatrix}a & b \\ c & d\end{bmatrix} \\ B = \begin{bmatrix}1 & 1 \\ 0 & 0\end{bmatrix} \\ AB = \begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 & 1 \\ 0 & 0\end{bmatrix} = \begin{bmatrix}a & a \\ c & c\end{bmatrix} $$

I am unable to see from this form whether the operation actually alters the number of solutions for the system. Is anyone able to advise me please?

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    Have you tried putting the matrix in row echelon form? – 2017-01-31
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    In the problem, I am not given an actual matrix. But I just defined an example matrix to try to see a relationship. The example matrix given for $ B $ above is in row echelon form. – 2017-01-31
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    No, the row echelon form for $AB$ is the question. – 2017-01-31
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    I am not given the matrix $AB$ either – 2017-01-31
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    When I said the matrix, I wasn't clear, but I meant the product matrix. – 2017-01-31
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    Since I only have variables to represent the values for $AB$, how do I put it in row-echelon form? – 2017-01-31
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    What do you know about the rank of a product of matrices? – 2017-01-31
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    I wasn't taught about rank. – 2017-02-01

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