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

The correct option is D $1$ if $a=-6$

$\left[\begin{array}{ccc}-1& 2& 5\\ 2& -4& a-4\\ 1& -2& a+1\end{array}\right]$

$R_2\to R_2+2R_1,R_3\to R_3+R_1$

$\left[\begin{array}{ccc}-1& 2& 5\\ 0& 0& a+6\\ 0& 0& a+6\end{array}\right]$

$R_3\to R_3-R_2$

$\left[\begin{array}{ccc}-1& 2& 5\\ 0& 0& a+6\\ 0& 0& 0\end{array}\right]$

Rank$=1$ if $a=-6$

Rank$=2$ if $a\ne -6$.