Answer
(a) A basis for the row space is given by
$$S=\{(4,0,2,3,1),(0,-1,1,-3),(0,0,6,-23,-5),(0,0,0,-1,0),(0,0,0,0,-3)\}.$$
(b) The rank of the matrix is $5$.
Work Step by Step
Let $A$ be given by $$
A=
\left[\begin{array}{rrrrr}{4} & {0} & {2} & {3} & {1} \\ {2} & {-1} & {2} & {0} & {1} \\ {5} & {2} & {2} & {1} & {-1} \\ {4} & {0} & {2} & {2} & {1} \\ {2} & {-2} & {0} & {0} & {1}\end{array}\right].
$$
The reduced form of $A$ is given by
$$ \left[ \begin {array}{ccccc} 4&0&2&3&1\\ 0&-2&2&-3&1\\ 0&0&6&-{23}&-5
\\ 0&0&0&-1&0\\ 0&0&0&0&-3
\end {array} \right]
$$
(a) A basis for the row space is given by
$$S=\{(4,0,2,3,1),(0,-1,1,-3),(0,0,6,-23,-5),(0,0,0,-1,0),(0,0,0,0,-3)\}.$$
(b) The rank of the matrix is $5$.
