This is a question about finding the rank of a matrix where one of the elements is an unknown variable.
I saw this online a few days ago and have been struggling to figure out how I find the rank of this matrix for all values of $a$: \begin{pmatrix} 3 & 1 & 1 & 4\\ a & 4 & 10 & 1\\ 1 & 7 & 17 & 3\\ 2 & 2 & 4 & 3\\ \end{pmatrix}
I know that the rank of a matrix can be deduced by figuring out how many independent column (or row) vectors the matrix has. Alternatively, the row-echelon-form of the matrix can be found and observed to see how many non-zero rows the $rref$ has, but the unknown variable is confusing me on doing both.
What is the best way to find the $rref$ or deduce independent row (or column) vectors? Any help would be greatly appreciated!
Here's the first step:
Rearranging some of the rows, we have $$\left[\begin{array}{cccc} 1&7&17&3 \\ 2&2&4&3 \\ 3&1&1&4 \\ a&4&10&1 \end{array} \right].$$
Then performing the row operations $R_2 \to R_2 - 2R_1$, $R_3 \to R_3 - 3R_1$ and $R_4 \to R_4 - aR_1$, we have $$\left[\begin{array}{cccc} 1&7&17&3 \\ 0&-12&-30&-3 \\ 0&-20&-50&-5 \\ 0&4-7a&10-17a&1-3a \end{array} \right].$$
Can you take it from here?