Question

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

• A. 1 if a = 6
• B. 2 if a = 1
• C. 3 if a = 2
• D. 1 if a = -6

Answer 1

Answer: (B), (D)

The correct options are
B 2 if a = 1
D 1 if a = -6

$A=\left|\begin{array}{ccc}-1& 2& 5\\ 2& -4& a-4\\ 1& -2& a+1\end{array}\right|=\left|\begin{array}{ccc}0& 0& a+6\\ 0& 0& -a-6\\ 1& -2& a+1\end{array}\right|$
(Operating $(R_1\to R_1+R_{3}and{R}_2\to R_2-2R_3)$
$=\left|\begin{array}{ccc}0& 0& 0\\ 0& 0& -a-6\\ 1& -2& a+1\end{array}\right|$(Operating $R_1\to R_1+R_2)$
When $a=-6,A=\left|\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& -2& -5\end{array}\right|$, $\therefore \rho \left(A\right)=1$
Where $\rho \left(A\right)$number of non-zero rows
When $a=6,A=\left|\begin{array}{ccc}0& 0& 0\\ 0& 0& -12\\ 1& -2& 7\end{array}\right|,\therefore \rho \left(A\right)=2$
When $a=1,A=\left|\begin{array}{ccc}0& 0& 0\\ 0& 0& -7\\ 1& -2& 2\end{array}\right|,\therefore \rho \left(A\right)=2$
When $a=2,A=\left|\begin{array}{ccc}0& 0& 0\\ 0& 0& -8\\ 1& -2& 2\end{array}\right|,\therefore \rho \left(A\right)=2$