How many solutions depending on the parameter (augmented matrix?)

Question

I have to find how many solutions have got the following equations, depending on p parameter?

$ \begin{bmatrix} 5 & p & 5 \\ 1 & 1 & 1 \\ p & p & 2 \end{bmatrix} $ $ \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = $ $ \begin{bmatrix} 5 \\ 1 \\ 2+p \end{bmatrix} $

Should I use Cramer's rule or maybe Gauss elimination? Thanks in advance.

Answer 1

Whenever the matrix is invertible, there is certain to be exactly one solution. So the first step is to find the $p$s for which the matrix is invertible.

It is easy to see that the determinant of the matrix is a polynomial of degree at most $2$ in $p$. This is not the zero polynomial (because the matrix clearly has full rank when $p=0$), so it can have at most two roots. It is also easy to see that the matrix in singular when $p$ is $5$ or $2$, so these must be the two roots.

Thus we know that there is exactly one solution whenever $p\notin\{2,5\}$.

You can then handle the $p=2$ and $p=5$ cases by explicit computation -- either Gaussian elimination or whichever clever shortcut you can see after you plug in $2$ or $5$ for $p$.