# 1. Determinants

by M. Bourne

Before we see how to use a matrix to solve a set of simultaneous equations, we learn about determinants.

A **determinant** is a square array of numbers (written
within a pair of vertical lines) which represents a certain sum
of products.

Below is an example of a 3 × 3 determinant (it has 3 rows and 3 columns).

`|(10,0,-3),(-2,-4,1),(3,0,2)|`

The result of multiplying out, then simplifying the elements of a determinant is a single number (a **scalar** quantity).

## Calculating a 2 × 2 Determinant

In general, we find the value of a 2 × 2 determinant with elements *a*,* b*,* c*,* d *as follows:

`|(a,b),(c,d)|=ad-cb`

We multiply the diagonals (top left × bottom right first), then subtract.

### Example 1

`|(4,1),(2,3)| =4xx3-2xx1 =12-2 = 10`

The final result is a single **number**.

We will see how to expand a 3 × 3 determinant below.

## Using Determinants to Solve Systems of Equations

We can solve a system of equations using determinants, but it becomes very tedious for large systems. We will only do 2 × 2 and 3 × 3 systems using determinants.

## Cramer's Rule

The solution (*x*, *y*) of the system

`a_1x+b_1y=c_1`

`a_2x+b_2y=c_2`

can be found using determinants:

`x=|(c_1,b_1),(c_2,b_2)|/|(a_1,b_1),(a_2,b_2)|`

`y=|(a_1,c_1),(a_2,c_2)|/|(a_1,b_1),(a_2,b_2)|`

### Example 2

Solve the system using Cramer's Rule:

x− 3y= 62

x+ 3y= 3

**3 ****×**** 3 Determinants**

A 3 × 3 determinant

`|(a_1, b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|`

can be evaluated in various ways.

We will use the method called "expansion by minors". But first, we need a definition.

## Cofactors

The 2 × 2 determinant

`|(b_2,c_2),(b_3,c_3)|`

is
called the **cofactor** of *a*_{1 }for the 3 × 3 determinant:

`|(a_1, b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|`

The cofactor is formed
from the elements that are not in the same row as *a*_{1} and not in the same column as *a*_{1}.

Similarly, the determinant

`|(b_1,c_1),(b_3,c_3)|`

is called the **cofactor** of *a*_{2}. It is formed from the
elements not in the same row as *a*_{2} and not in the same column as *a*_{2}.

We continue the pattern for the cofactor of *a*_{3}.

## Expansion by Minors

We evaluate our 3 × 3 determinant using expansion by minors. This involves multiplying the **elements** in the first column of the determinant by the **cofactors** of those elements. We subtract the middle product and add the final product.

`|(a_1, b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=a_1|(b_2,c_2),(b_3,c_3)|-a_2|(b_1,c_1),(b_3,c_3)|+a_3|(b_1,c_1),(b_2,c_2)|`

**Note** that we are working down the first column and multiplying by the cofactor of each element.

### Example 3

Evaluate

`|(-2,3,-1),(5,-1,4),(4,-8,2)|`

(You can explore what this example is really asking in this 3D interactive systems of equations applet.)

## Cramer's Rule to Solve 3 × 3 Systems of Linear Equations

We can solve the general system of equations,

a_{1}x+b_{1}y+c_{1}z=d_{1}

a_{2}x+b_{2}y+c_{2}z=d_{2}a_{3}x+b_{3}y+c_{3}z=d_{3}

by using the determinants:

`x=|(d_1, b_1,c_1),(d_2,b_2,c_2),(d_3,b_3,c_3)|/Delta`

`y=|(a_1, d_1,c_1),(a_2,d_2,c_2),(a_3,d_3,c_3)|/Delta`

`z=|(a_1, b_1,d_1),(a_2,b_2,d_2),(a_3,b_3,d_3)|/Delta`

where

`Delta=|(a_1, b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|`

### Example 4

Solve, using Cramer's Rule:

2

x+ 3y+z= 2−

x+ 2y+ 3z= −1−3

x− 3y+z= 0

## Determinant Exercises

1. Evaluate by expansion of minors:

`|(10,0,-3),(-2,-4,1),(3,0,2)|`

2. Solve the system by use of determinants:

x+ 3y+z= 42

x− 6y −3z= 104

x− 9y+ 3z= 4

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