Thread: Row space of a matrix?

1. Row space of a matrix?

How do I prove:

$row(A^2)\subset row(A)$, for a matrix A

2. each row of A^2 can be shown to be linear combination of rows of A. (look at the definition of matrix multiplication). Hence row space of A^2 is a subset of row space of A.

Not sure if there is another more conceptual argument.
Thanks