Consider the linear system of equations AX = Y
where
X is a n x 1 matrix of variables,
Y is a n x 1 matrix of constants, and
A is an n x n matrix of coefficients.
Provided A is not a singular matrix, A has an inverse, A-1, an n x n matrix.
Premultiplying by A-1 gives A-1AX = A-1Y or X = A-1Y, the solution to the linear system.
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Many problems in economics can be modelled by a system of linear equations: equalities r inequalities. Such systems are best solved using matrix algebra.
There are more than two methods, and of these, matrix inversion is probably the easiest for solving systems of linear equations in several unknowns.
The concept of systems of linear equations dates back to ancient civilizations such as Babylonians and Egyptians. However, the systematic study and formalization of solving systems of linear equations is attributed to the ancient Greek mathematician Euclid, who introduced the method of substitution and elimination in his work "Elements." Later mathematicians such as Gauss and Cramer made significant contributions to the theory and methods of solving systems of linear equations.
When the matrix of coefficients is singular.
Solving linear systems means to solve linear equations and inequalities. Then to graph it and describing it by statical statements.