3) Find the rank of the matrix A [1 3 1 -2 -3 1 A= 4 3 -1 -4 23 -4 -7 -3 3 8 1 -7 -81

Consider the matrix 

$A=\left[\begin{array}{cccc}1& 3& 1& -2\\ 1& 4& 3& -1\\ 2& 3& -4& -7\\ 3& 8& 1& -7\end{array}\begin{array}{c}-3\\ -4\\ -3\\ -8\end{array}\right]$

$A=\left[\begin{array}{cccc}1& 3& 1& -2\\ 1& 4& 3& -1\\ 2& 3& -4& -7\\ 3& 8& 1& -7\end{array}\begin{array}{c}-3\\ -4\\ -3\\ -8\end{array}\right]$

Apply row operation $R_2\to R_2-R_1,R_3\to R_3-2R_1,\text{ and }{R}_4\to R_4-3R_1$

$A=\left[\begin{array}{cccc}1& 3& 1& -2\\ 0& 1& 2& 1\\ 0& -3& -6& -3\\ 0& -1& -2& -1\end{array}\begin{array}{c}-3\\ -1\\ 3\\ 1\end{array}\right]$

Apply row operation $R_3\to R_3+3R_2$ 

$A=\left[\begin{array}{cccc}1& 3& 1& -2\\ 0& 1& 2& 1\\ 0& 0& 0& 0\\ 0& -1& -2& -1\end{array}\begin{array}{c}-3\\ -1\\ 0\\ 1\end{array}\right]$

Apply row operation $R_4\to R_4+R_1$

$A=\left[\begin{array}{cccc}1& 3& 1& -2\\ 0& 1& 2& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\begin{array}{c}-3\\ -1\\ 0\\ 0\end{array}\right]$

There are two pivots, hence there are two linear independent rows.

Rank of A =2