Summary Simultaneous Linear Equations

Simultaneous Linear Equations
This lesson explains how to use matrix methods to (1) represent a system of linear equations compactly and (2) solve simulataneous
linear equations efficiently.


How to Represent a System of Linear Equations In Matrix Form
Suppose you have n linear equations with n unknowns. Using ordinary algebra, those equations might be expressed as:

A11x1 + A12x2 + A13x3 + . . . + A1nxn = y1
A21x1 + A22x2 + A23x3 + . . . + A2nxn = y2
A31x1 + A32x2 + A33x3 + . . . + A3nxn = y3
...
An1x1 + An2x2 + An3x3 + . . . + Annxn = yn

where

xj is an unknown value
Aij is the known coefficient of xj in equation i
yj is a known quantity in equation j

This set of equations can be expressed compactly in matrix form as follows:

Ax = y

where

x is an n x 1 column vector of unknown values x1, x2, . . . , xn
A is an n x n matrix of the known coefficients Aij
y is an n x 1 column vector of known values y1, y2, . . . , yn


How to Solve Simultaneous Linear Equations Using Matrix Methods

Here is how to solve a system of n linear equations in n unknowns, using matrix methods.



Express the set of n linear equations compactly in matrix form.

Ax = y




Premultiply both sides of the equation by A-1, the inverse of A.

A-1Ax = A-1y




Since A-1Ax = Ix = x, we know the following.

x = A-1y