What can be the rank of the following matrix, where $a$ is a parameter?

Question

The matrix is $ A = \begin{pmatrix} 1 & 1 & -1 & 2 \\ a & 1 & 1 & 1 \\ 1 & -1 & 3 & -3 \\ 4 & 2 & 0 & a \end{pmatrix}$

I have "tested" the matrix and I always get that the rank of the matrix is 4. How can I show that after the Gauss Elimination the number of linearly independent rows will be always 4?

Answer 1

The rank is equal to the dimensión of the row space.

since the determinant is $-2(a-3)^2$ we conclude the rank is $4$ is $a\neq 2$.

Otherwise rows $2$ and $4$ are in the span of columns $1$ and $3$, so the rank is $3$