Answer
$nullity (A)=3$
Work Step by Step
We are given $A=\begin{bmatrix}
1 & 3 & -3 & 2 & 5\\
-4 & -12 & 12 & -8 & -20\\
0 & 0 & 0 & 0 & 0\\
1 & 3 & -3 &2 & 6
\end{bmatrix}$
In this problem, A is a $5 \times 4$ matrix, and so, in the Rank-Nullity Theorem, $n = 5$. Further, from the foregoing row-echelon form of the augmented matrix of the system
Ax = 0, we see that $rank(A) = 2$.
Hence:
$$rank (A)+nullity(A)=n \\
nullity (A)=n-rank (A)=5-2=3$$
