The jargon is sometimes confusing for me. So here is a cheat sheet to refer to if you forget:

• Non-trivial solution: where not all the coefficients are equal to zero (i.e. there is at least one non-zero coefficient).
• Homogeneous: where all the equations are equal to zero.

Elementary matrices

These basically involve the manipulation of a unit matrix somehow. i.e.

• Unit matrix multiplied by a number c.
• Where you swap two rows of a unit matrix
• Where you have a unit matrix and one other number in the matrix (apart from a diagonal element) equal to a number c, where c is not zero.