Question

What is matrix algebra used for?

Answer 1

Two relatively simple examples of the use of matrix algebra are
`color(white)("XXXX")`Cramer's Rule and
`color(white)("XXXX")`Markov Chains

Explanation:

I certainly remember being subjected to a year of matrix algebra (by a teacher with no math background) and having no indication that it was of anything but theoretical use, so I can empathize with this question.

I won't get into the details (unless asked separately) but here are 2 applications:

Cramer's Rule
Given a system of linear equations, the value of a variable can be determined by:
dividing the value of the variable replaced determinant by of the value of the coefficient determinant.
For example, given
#color(white)("XXXX"){: (2x, +5y, -3z, =, color(red)(7)), (5x, -3y, 1z, =, color(red)(9)), (3x, +2y, +4z, =, color(red)(6)) :}#

`x = | {: (color(red)(7), color(white)("XX"),5, color(white)("XX"),-3), (color(red)(9),color(white)("XX"), -3, color(white)("XX"),1), (color(red)(6),color(white)("XX"), 2,color(white)("XX"),4) :}|/| {: (2,color(white)("XX"), 5, color(white)("XX"),-3), (5, color(white)("XX"),-3, color(white)("XX"),1), (3,color(white)("XX"), 2,color(white)("XX"),4) :}|`

Markov Chains
Given a square matrix `M` with elements `a_(ij)`
where the value of `a_(ij)` is the probability of a change from a state `i` to a state `j` in a single cycle

`M^k` gives a matrix of the probabilities of moving from a state `i` to a state `j` after `k` cycles.