Determinants of Matrices
Determinants of Matrices
Matrices are used to solve simultaneous equations.
A determinant of a matrix represents a single number. It can be used to solve a system of simultaneous equations. For example, the value of a 2 x 2 determinant would be found as follows:
Determinants can be used to solve a system of two equations.
ax + by = M
cx + dy = N
In this system, let D represent the matrix formed by the variables' coefficients, with each variable in its own column. D_{x} represents the same matrix, except the x column is replaced with the solution column (the one on the right of the equal sign). Likewise, D_{y} represents the matrix with the y column replaced with the solution column.
The solution of the system can be found using the following calculations with the determinants of these matrices:
x = y =
This is known as Cramer's Rule.
Example
Solve the following system using determinants:
4x - 3y = 92x + 7y = 13
Set up the matrices as specified by Cramer's Rule and find their determinants:
|D| =
|D_{x}| =
|D_{y}| =
Therefore,
x = ^{102}/_{34} = 3y = ^{34}/_{34} = 1