Question

If the rank of the matrix $\left[\begin{array}{ccc}4& 2& 1-x\\ 5& k& 1\\ 6& 3& 1+x\end{array}\right]$ is $2$ then

• A. $k=\frac{5}{2},x=\frac{1}{5}$
• B. $k=\frac{5}{2},x\ne \frac{1}{5}$
• C. $k=\frac{1}{5},x=\frac{5}{2}$
• D. none of these

Answer 1

Answer: (B)

$k=\frac{5}{2},x\ne \frac{1}{5}$

rank of the matrix $\left[\begin{array}{ccc}4& 2& 1-x\\ 5& k& 1\\ 6& 3& 1+x\end{array}\right]$ is $2$

Note: Rank of a matrix is nothing but, no of non-zero rows of a matrix in its echelon form.
applying $R_3\to R_1+R_3$

$\Rightarrow \left[\begin{array}{ccc}4& 2& 1-x\\ 5& k& 1\\ 10& 5& 2\end{array}\right]$

applying $R_3\to R_3-2R_2$

$\Rightarrow \left[\begin{array}{ccc}4& 2& 1-x\\ 5& k& 1\\ 0& 5-2k& 0\end{array}\right]$

applying $R_2\to R_2-\frac{5}{4}{R}_1$

$\Rightarrow \left[\begin{array}{ccc}4& 2& 1-x\\ 0& k-\frac{5}{2}& \frac{5x-1}{4}\\ 0& 5-2k& 0\end{array}\right]$

rank is 2

$\therefore k=\frac{5}{2}andx\ne \frac{1}{5}$