Unlikely. Lots of mathematicians' hearts jump every time someone finds a slick algebraic approach to some messy combinatorics, but it's not a very frequent occurrence (I think of finding such approaches as the main part of my job).

In a typical discrete maths course, the main "arbitrage" you get from algebra is the "method" of generating functions, which is a technique for dealing with integer sequences and counting problems using polynomials. If you like matrices, you can encode polynomials as Toeplitz matrices, so you can view products of polynomials as matrix products. But generating functions are not a panacea for counting, and you usually have to seed the ground with some combinatorial results before you can water it with generating functions. Also, there are some neat uses of matrices in understanding Pascal's triangle ( http://www-math.mit.edu/~gs/papers/pascal-work.pdf ) and other sequences.

In graph theory, the matrix-tree theorem lets you count spanning trees using a determinant, and the Dijkstra algorithm for path finding can be regarded as a matrix power over the max-plus semiring. But that's about all you get out of linear algebra in a typical first discrete maths course. Matrices become more useful if you go deeper into graphs or into counting, but never take center stage.

[And before the next question comes: I haven't seen many monads used in discrete maths either.]