Matrices are considered very important in mathematics. What are some examples of applications of matrices to real world problems that would be understandable by a layman?
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If you have system of nonlinear equations and linearize them, you get a linear system, which can be represented by a matrix.
So, matrices are relevant whenever you can model the real world with equations, which is very often.
One would be hard pressed to find a tool in mathematics that is more widely used for real world applications than matrices and linear algebra.
These are only a few examples, but let me assure you that nearly any quantitative line of work will make extensive use of matrix equations.