# How do I know if a matrix is irreducible?

My course at university mainly works with 3x3 matrices. We are asked to put them in reduced echelon form which is the easy part, however I come across many matrices that I cannot seem to reduce into echelon form, I do not know if it is irreducible or I am just taking the wrong approach to reduce the matrix.

I was told if the matrix is able to be put into upper triangular form it is reducible but again, I am not sure if I am doing the question wrong or in actual fact that matrix is irreducible.

Any help would be much appreciated!