Answer
$A$ has no inverse or is non-invertible (or singular).
Work Step by Step
To find $A^{-1}$, we have
$$\left[ A \ \ I \right]= \left[ \begin {array}{cccccc} 3&2&5&1&0&0\\ 2&2&4&0
&1&0\\ -4&4&0&0&0&1\end {array} \right]
.
$$
Using Gauss-Jordan elimination, we get the row-reduced echelon form as follows
$$ \left[ \begin {array}{cccccc} 1&0&1&0&\frac{1}{4}&-\frac{1}{8}\\ 0&
1&1&0&\frac{1}{4}&\frac{1}{8}\\ 0&0&0&1&-\frac{5}{4}&\frac{1}{8}\end {array}
\right]
.
$$
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).