If we let $A$ be an $m × n$ matrix. Suppose that the rows of $A$ are linearly independent.

Question

If we let $A$ be an $m × n$ matrix. Suppose that the rows of $A$ are linearly independent.
(a) Is $A^TA$ invertible? Explain.
(b) Is $AA^T$ invertible? Explain.

So if the rows of $A$ are linearly independent then $Ax=0$. But how do I go about proving a and b?

Answer 1

I assume that you are working over the reals.

Assume the rows of $A$ are linearly independent. That means that the only solution to $xA = 0$ is $x = 0$ where $x$ is a row vector of length $m$.

Now assume that $AA^T$ is not invertible. Then there exists a row vector $v$ such that $vAA^T = 0$ (easy to see if we are over the reals or any other field, a little harder over arbitrary rings). But then also $vAA^Tv^T = 0$. Setting $x := vA$ we thus have $xx^T = 0$, which might ring some bells and then lead to a contradiction.