Determinants of Matrices

Determinants of Matrices

Matrices are used to solve simultaneous equations.

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.  Dx represents the same matrix, except the x column is replaced with the solution column (the one on the right of the equal sign).  Likewise, Dy 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 = 9
2x + 7y = 13

Set up the matrices as specified by Cramer's Rule and find their determinants:

|D| =

|Dx| =

|Dy| =

Therefore,
x = 102/34 = 3
y = 34/34 = 1