### Instruction: Let $\mathbf{A} = \begin{pmatrix} 1 & 5 \\ 1 & 3 \end{pmatrix}.$  Find all real numbers $x$ such that that matrix $\mathbf{A} - x \mathbf{I}$ is not invertible.  Enter all the solutions, separated by commas.### Response: To solve this problem, I need to recall what it means for a matrix to be invertible or not.
A matrix is invertible if and only if its determinant is not zero.
The determinant of a 2 by 2 matrix is the product of the diagonal entries minus the product of the off-diagonal entries.
So the determinant of $\mathbf{A} - x \mathbf{I}$ is $(1-x)(3-x) - 5 \cdot 1 = x^2 - 4x - 2$.
This is a quadratic polynomial, and I can use the quadratic formula to find its roots.
The roots are $x = \frac{4 \pm \sqrt{16 + 8}}{2} = 2 \pm \sqrt{6}$.
These are the values of $x$ that make the determinant zero, and therefore make the matrix not invertible.
I need to enter them in ascending order, separated by commas.