Question

The rank of the matrix$A=\left[\begin{array}{cccc}1& 2& -1& 3\\ 4& 1& 2& 1\\ 3& -1& 1& 2\\ 1& 2& 0& 1\end{array}\right]$ is

• A. 3
• B. 1
• C. 2
• D. 4

Answer 1

The correct option is A 3

Order of matrix is $(4\times 4)$. So, Row Echelon form should be preferred over Determinant method.

$A={\left[\begin{array}{cccc}1& 2& -1& 3\\ 4& 1& 2& 1\\ 3& -1& 1& 2\\ 1& 2& 0& 1\end{array}\right]}^{R_2\to R_2-4R_1{R}_3\to R_3-3R_1{R}_4\to R_4-R_1}$

$A=\left[\begin{array}{cccc}1& 2& -1& 3\\ 0& -7& 6& -11\\ 0& -7& 4& -7\\ 0& 0& 1& -2\end{array}\right]R_3\to R_3-R_2$

$A=\left[\begin{array}{cccc}1& 2& -1& 3\\ 0& -7& 6& -11\\ 0& 0& -2& 4\\ 0& 0& 1& -2\end{array}\right]R_4\to 2R_4+R_3$

$A=\left[\begin{array}{cccc}1& 2& -1& 3\\ 0& -7& 6& -11\\ 0& 0& -2& 4\\ 0& 0& 0& 0\end{array}\right]\left(Echelonform\right)$

$\because$$\rho$(A) = 3 (No. of non zero rows in Echelon form)