Let $a \; \in \; \mathbb{R}$. Find the rank of the matrix
$$\begin{pmatrix} -a&1&2&3&1\\ 1&-a&3&2&1\\ 2&3&-a&1&1\\ 3&2&1&-a&1 \end{pmatrix}$$
This is one of my exercises. I tried to use some elementary operations to transform this matrix to RRE form, but it seems like it is really complicated to complete. Help me, please. Thanks!
On
Hint: Since row rank equals column rank, the rank of the matrix is at most $4$. To prove that the rank is $4$, it suffices to compute the determinant of two $4\times 4$ minors and check that they don't have common factor as polynomials in ${\mathbb R}[a]$. For example, the matrix consisting of the first four column vectors has determinant $a(a+4)(a+2)(a-6)$. Then either you can compute one more determinant and check for common factor, or work on the four cases $a=0,-4,-2,6$ to see that the resulting matrix indeed has rank $4$. Note that if $a$ is none of the above values, the rank will be $4$.
Not sure where you are getting stuck. Apply Gaussian elimination to convert to reduced echelon form and constraints on $a$ will be obvious from there.