To simplify notation and to get a better overview of the data.
Anything you can do with matrices is something you can do without (like linear maps for example), but it more annoying and less straightforward. Matrices really allow you to use algorithms. This is very handy for computing.
Computer graphics, for instance. To me matrices represent systems of equations - that's what I use a matrix for in the first place - ok you might be used to solving a 3 variable system with some substitutions and get the answer. In practice, let's say you build a bridge and want to know the properties of it. There are So many variables to the system that you will finish your calculations by the end of 2050. Instead you use matrices and apply something called the iteration method as long until you reach a desired accuracy for the results and save a heck of a long time in the process.
Matrices are used every where. Many real world problems are describe, not by 1 single equation, but instead by systems of multiple equations. Matrices are the fundamental tool used in organizing and solving these systems of equations.
Ecologist use matrices to explore the population dynamics of predator and preys (like foxes and rabbits)
Chemists use matrices to track the concentrations of reactants and products in a chemical reactor.
Physicist, mathematicians, engineers, computer scientists, economists, climate scientist, geologist, etc all matrices to solve a variety of different problems.
There isn't one typical problem where matrices are used, just like there isn't one typical problem where multiplication is used.
A lot of the uses of matrices are hard for a newcomer to understand, but if you've seen some basic algebra, you've probably solved systems of linear equations. If you have 2 equations and 2 unknowns, you don't really need matrices to solve it. But what if you have 100 equations and 100 unknowns? Using matrices makes it a little less messy. Matrices also lead to a more elegant way of thinking about it, which is hard to appreciate unless you delve into the ideas of linear algebra. These ideas can deepen your understanding of a quite a number of things, and matrices are part of that story. It goes beyond just using matrices directly to compute this or that.
Here's one good application. Keeping track of different stresses and strains in materials: