HCU MCA 2013 Paper

(c) Given matrix,
${\left[\begin{array}{ccc}-1& 4& 2\\ 1& 3& 2\\ -2& 1& 0\\ 2& 6& 4\end{array}\right]}_{4\times 3}$
Number of columns (n) = 3
Apply operation, ${\mathrm{R}}_1\to {\mathrm{R}}_2$
A ~$\left[\begin{array}{ccc}1& 3& 2\\ -1& 4& 2\\ -2& 1& 0\\ 2& 6& 4\end{array}\right]$
Using operations ${\mathrm{R}}_2\to {\mathrm{R}}_1+{\mathrm{R}}_2,{\mathrm{R}}_3\to {\mathrm{R}}_3+2{\mathrm{R}}_1$ and ${\mathrm{R}}_4\to {\mathrm{R}}_4-2{\mathrm{R}}_1;$
A ~$\left[\begin{array}{ccc}1& 3& 2\\ 0& 7& 4\\ 0& 7& 4\\ 0& 0& 0\end{array}\right]$
Again, apply operation, ${\mathrm{R}}_3\to {\mathrm{R}}_3-{\mathrm{R}}_2$
A ~$\left[\begin{array}{ccc}1& 3& 2\\ 0& 7& 4\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
By Echlon form, Rank of A = f(A)
$=$Number of non-zero rows$$ 2
∴ Nullity A = n - f(A) = 3 - 2 = 1.