[Math] Find rank and nullity of a matrix.

Question

To find:
rank $A$ and nullity $A$ for
$$A=\begin{pmatrix}
0 &0 &0 \\
0 & 0.5&-0.5 \\
0&-0.5 & 0.5
\end{pmatrix}$$
I know the nullity refers to the number of free variables in the matrix and the rank refers to the $dim(columnspace)$; where to from here?

Answer 1

More Generally. First you are going to want to set this matrix up as an Augmented Matrix where $Ax=0$.

$1)$ To find the rank, simply put the Matrix in REF or RREF

$\left[\begin{array}{ccc|c} 0 & 0 & 0 &0 \\ 0 & 0.5 & -0.5 & 0 \\ 0 & -0.5 & 0.5 & 0 \end{array}\right] \longrightarrow RREF \longrightarrow \left[\begin{array}{ccc|c} 0 & 0 & 0 &0\\ 0 & 0.5 & -0.5 & 0\\ 0 & 0 & 0 & 0 \end{array}\right] $

Seeing that we only have one leading variable we can now say that the rank is 1.

$2)$ To find nullity of the matrix simply subtract the rank of our Matrix from the total number of columns.

So:

Null (A)=3 - 1=2

Hope this is helpful.