Diagonalisable or not?

Question

Let $$A = \begin{pmatrix}a & b\\c & d\end{pmatrix}.$$

Show that

1) $A$ is diagonalisable if $(a - d)^2 + 4 bc > 0$

2) $A$ is not diagonalisable if $(a - d)^2 + 4 bc < 0$

Answer 1

Hint: Look at the characteristic polynomial and see what those conditions imply.

Answer 2

The characteristic polynomial is

$$\det(xI-A)=\begin{vmatrix}x-a&-b\\-c&x-d\end{vmatrix}=x^2-(a+d)x+(ad-bc)$$

The above quadratic's discriminant is

$$\Delta=(a+d)^2-4(ad-bc)=(a-d)^2+4bc$$

Now, what does having positive discriminant imply for the roots of a quadratic? And what does having two different eigenvalues imply about diagonalizing (or not) a $\;2\times2\;$ matrix? And what if no eigenvalues exist?