Quick way to check if a matrix is diagonalizable.

Firstly make sure you are aware of the conditions of Diagonalizable matrix.

In a multiple choice setting as you described the worst case scenario would be for you to diagonalize each one and see if it's eigenvalues meet the necessary conditions.

However, as mentioned here:

A matrix is diagonalizable if and only if for each eigenvalue the dimension of the eigenspace is equal to the multiplicity of the eigenvalue.

Meaning, if you find matrices with distinct eigenvalues (multiplicity = 1) you should quickly identify those as diagonizable.

It also depends on how tricky your exam is. For instance if one of the choices is not square you can count it out immediately. On the other hand, they could give you several cases where you have eigenvalues of multiplicity greater than 1 forcing you to double check if the dimension of the eigenspace is equal to their multiplicity.

Again, depending on the complexity of the matrices given, there is no way to really spot-check this unless you're REALLY good at doing this all in your head.