Discussion Forum

The rank of the matrix $\begin{bmatrix}-1 & 2 & 5\\2 & -4 & a-4\\ 1 & -2 & a+1 \end{bmatrix}$ is

Answer:

Explanation:

Let

$A=\begin{bmatrix}-1 & 2 & 5\\2 & -4 & a-4\\ 1 & -2 & a+1 \end{bmatrix}\sim\begin{bmatrix}-1 & 2 & 5\\0 & 0 & a+6\\ 0 & 0 & a+6 \end{bmatrix}$

[R₂ → R₂ + 2R₁, R₃ → R₃ +R₁]

Clearly rank of A is 1 if a = -6

Also, for a= 1, | A | = $\begin{vmatrix}-1 & 2 & 5\\2 & -4 & -3\\ 1 & -2 & 2 \end{vmatrix}=0$

and $\begin{vmatrix}2 & 5 \\-4 & -3 \end{vmatrix}= -6+20=14\neq0$

.'. rank of A is 2 if a = 1