This problem is in Trefethen'book Numerical Linear Algebra
Suppose the $m\times n$ matrix $A$ has the form
$A=\begin{pmatrix}A_1\\A_2 \end{pmatrix}$
where $A_1$ is a nonsingular matrix of dimension $n\times n$ and $A_2$ is an arbitrary matrix of dimension $(m-n)\times n$. Prove $\|A^+\|_2\leq \|A_1^{-1}\|_2$. where $A^+=(A^*A)^{-1}A^*$