Answer
$A$ has no inverse or is non-invertible (or singular).
Work Step by Step
To find $A^{-1}$, we have
$$\left[ A \ \ I \right]= \left[ \begin {array}{cccccc} 1&2&-1&1&0&0\\ 3&7&-
10&0&1&0\\ 7&16&-21&0&0&1\end {array} \right]
.
$$
Using Gauss-Jordan elimination, we get the row-reduced echelon form as follows
$$ \left[ \begin {array}{cccccc} 1&0&13&0&-16&7\\ 0&1&
-7&0&7&-3\\ 0&0&0&1&2&-1\end {array} \right]
.
$$
Note that the portion of the matrix $A$ has a row of zeros. So it is not possible to rewrite the matrix $\left[ A \ \ I \right] $ in the form $\left[I \ \ A^{-1} \right]$. This means that has no inverse, or is non-invertible (or singular).