Notes for Homework 8, due March 28

Notes on determinants

A 2x2 matrix is an array of four numbers put in two rows and two columns, such as

| a c |
| b d |

The determinant of this matrix is ad - bc, the product of the main diagonal minus the product of the opposite diagonal. If you use Kramer's rule to solve a pair of simultaneous equations, you will need to know how to calculate determinants.

For example, let's consider the following set of equations.

3x - 2y = 7
2x + 4y = 12

As an augmented matrix , these equations become

|  3  -2  : 7 |
|  2   4  : 12|

Matrix_1 is

| 3 -2 |
| 2  4 |

The determinant is 3(4) - 2(-2) = 12 - -4 = 16
 

Matrix_x is

| 7 -2 |
| 12  4 |

The determinant is 7(4) - 12(-2) = 28 - -24 = 52
  Matrix_y is

| 3  7  |
| 2 12 |

The determinant is 3(12) - 2(7) = 36 - 14 = 22

From here, Kramer's rule then has x = 52/16 = 13/4 or 3 1/4. The value for y = 22/16 = 11/8 = 1 3/8
 
  Notes on solving simultaneous equations