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 if $a=1$ and if $a=-6$

• A. 2
• B. 1
• C. 3
• D. 0.0

Answer 1

Answer: (A), (B)

The correct options are
A 2
B 1

$A=\left[\begin{array}{ccc}-1& 2& 5\\ 2& -4& a-4\\ 1& -2& a+1\end{array}\right]=\left[\begin{array}{ccc}0& 0& a+6\\ 0& 0& -a-6\\ 1& -2& a+1\end{array}\right]$
$[R_1\to R_1+R_3$ and $R_2\to R_2-2R_3]$

$=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -a-6\\ 1& -2& a+1\end{array}\right]$
$[R_1\to R_1+R_2]$

When $a=-6,A=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& -2& -5\end{array}\right]$
$\Rightarrow \rho \left(A\right)=1$, where $\rho \left(A\right)$ is the number of non-zero rows which gives us the rank.

When $a=6,A=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -12\\ 1& -2& 7\end{array}\right]$
$\Rightarrow \rho \left(A\right)=2$

When $a=1,A=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -7\\ 1& -2& 2\end{array}\right]$
$\Rightarrow \rho \left(A\right)=2$

When $a=2,A=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -8\\ 1& -2& 2\end{array}\right]$
$\Rightarrow \rho \left(A\right)=2$