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
Thus, as long as the inverse A-1 exists, we
can solve for x, the vector of unknown values. If the
inverse does not exist, the set of equations does not have a unique
solution.
Solving Simultaneous Linear Equations: An Example
To illustrate how to solve simultaneous linear equations using matrix methods,
consider the following system of equations.
x1 + 2x2 + 2x3 = 1
2x1 + 2x2 + 2x3 = 2
2x1 + 2x2 + x3 = 3
We want to solve for the unknown quantities: x1, x2,
and x3.
- Our first step is to express these equations
in matrix form as Ax = y.
- And finally, since
A-1Ax
= Ix = x,
we know that x = A-1y.
Thus,
Thus, we have solved for the unknown quantities:
x1 = 1, x2 = 1, and x3 = -1.
Test Your Understanding
Problem 1
Consider the following system of linear equations.
3x1 + x2 = 3
9x1 + 4x2 = 6
Using matrix methods, solve for the unknown quantities:
x1and x2.
Solution
Our solution involves a three-step process.
- The first step is to express these equations
in matrix form as Ax = y.
- And finally, since
A-1Ax
= Ix = x,
we know the following.
Thus, we have solved for the unknown quantities:
x1 = 2 and x2 = -3.