[Math] Let $L$ denotes the set of all primes $p$ such that the following matrix is invertible

Question

I am stuck on the following problem:

Let $L$ denotes the set of all primes $p$ such that the following matrix is invertible when considered as a matrix with entries in $\Bbb Z/p \Bbb Z$ .

$A=\begin{pmatrix}
1 &2 &0 \\
0 &3 &-1 \\
-2 &0 &2
\end{pmatrix}$

Then how can I verify whether the following statements are true/false?

1. $L$ contains all the prime numbers greater than $10$

2. $L$ contains all the prime numbers other than $2$ and $5$

3. $L$ contains all the prime numbers

4. $L$ contains all the odd prime numbers.

Can someone give explanation? Thanks in advance for your time.

Answer 1

Hint: a square matrix over any field is invertible iff its determinant is nonzero (as a member of the field).