Determinants of Matrices


Determinants of Matrices

Matrices are used to solve simultaneous equations.  In some cases, you don't have to use row operations.


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.

       

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 = = 3

y = = 1



Practice

1.  Find the determinants of the following matrices.

a. 


b. 


c. 


2.  Find |D|, |Dx|, |Dy|, and the solutions to the following system:

2x − 3y = 16
5x + 4y = 17













Answers

1.  a. 14  b. -11  c. 13

2.  |D| = 23;  |Dx| = 115;  |Dy| = -46;  x = 5;  y = -2