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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?

Rushabh Mehta
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lirui xiao
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  • https://math.stackexchange.com/q/1181271/ , https://math.stackexchange.com/q/1840801 – Trevor Gunn Dec 08 '19 at 02:57
  • Not sure what the first column has to do with anything. If $N < p + 1$ or $X$ has two identical columns then it has a non-trivial null space. Then if $Xv = 0$ we have $X^TXv = 0$ so $X^TX$ cannot be invertible. – Trevor Gunn Dec 08 '19 at 02:58

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