Show that the group $\mathrm{GL}_2(\mathbb{F}_5)$ has order 480. By defining a suitable homomorphism from $\mathrm{GL}_2(\mathbb{F}_5)$ to another group, which should be specified, show that the order of $\mathrm{SL}_2(\mathbb{F}_5)$ is 120. Find a subgroup of $\mathrm{GL}_2(\mathbb{F}_5)$ of index 2.
For the first part I actually can show the order 480. First I find the order of $\mathrm{M}_2(\mathbb{F}_5)$ , $5^4=625$, then I discard all the singular matrices then I have 480. But find all the cases that the matrices are singular is tedious and not easy to ensure every case is considered in the field. I want to find a simpler and more effective method to approach this.