Matrices are studied in linear algebra. One of their applications is the simple solution of systems of linear equations. An inverse matrix, when it exists, solves these systems through either the mathematics of manipulating these matrices or through numerical analysis by a software program, such as Matlab.
In Matlab, a matrix can be entered into the computer so the process of finding the inverse matrix is handled automatically.
First, the expressions that are used for such a system of equations in matrix notation are in the form Ax = B, where A and B are matrices, and x is a matrix of variables for which one is seeking a solution. Also, remember that the multiplication of matrices is not commutative, so operations are denoted "left-" and "right-" to indicate the type of multiplication needed.
First, left-multiply the system by the inverse matrix to A (denoted invA) to give:
invA * (Ax) = invA * B.
Fortunately, multiplication of matrices is associative, so regrouping provides:
(invA * A) x = invA * B => x = invA * B.
It should be noted that there are more efficient ways to accomplish this solution in some cases, but there also may be cases where this is the only method available.