Suppose that the first column of $X \in R^{N \times (p+1)}$ is full of $1$s. Show that in the following cases, there exists no inverse matrices for $X^TX$. (a). $N < p+1$ (b). $N \ge p+1$ and two columns of $X$ are the same.

So far, I know 1. for a matrix to be invertible, it has to be a square matrix. 2.$A^{-1}A = I = AA^{-1}$ for an invertible matrix. I do not understand the statement "the first column is full of $1$" How should this information be used to solve the problem? Could someone explain the idea?