# If a matrix A is injective then AAt is invertible

1. Oct 15, 2009

1. The problem statement, all variables and given/known data
If a matrix, A (nxm) is monic (or epic) then is $$A^tA (or AA^t)$$ is invertible?

2. Relevant equations
T is monic if for any matrices B,C: BT = BT => B=C.
S is invertible if there exists U s.t. US = SU = $$I_n$$

3. The attempt at a solution
Since A is monic it must preserve n; i.e. assume n < m, so that A has n independent rows (is that allowed), and we view A as an function of natural numbers, $$n \stackrel{A}{\rightarrow} m$$. Then can we say AB where B is m * k, means that AB has n indepenent rows as well?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution