Use Cramer's Rule to express the solution of a system in Determinant Form

Determinant

An n^th order determinant is a square nxn array or pattern of numbers, comprised of rows and columns.

2nd order or 2x2 determinant:
row 1, row 2 ; column 1, column 2

2	3
5	-2

3nd order or 3x3 determinant:
row 1, row 2, row 3 ; column 1, column 2, column 3

-4	5	8
2	1	7
-8	0	15

• Each entry or number in the determinant is referred to as an element of the determinant

• A determinant evaluates to a number (if all the elements in that determinant are numeric)

Calculating or Evaluating a second-order determinant or 2x2 determinant
Method A - "criss-cross" method

a	b
c -	d +

= a d - b c



Method B - "co factor expansion" method

ex. evaluate  ∆ =

2	4
10	-3

= (2) (-3) - (4) (10) = - 46

∆ - delta, often used to symbolize a given determinant.

ex. evaluate ∆ =

4	10
2	7

= (4) (7) - (2) (10) = 8

Calculating or Evaluating a third-order determinant or 3x3 determinant



Method A - "criss-cross" method

	a	b	c	a	b
	d	e	f	d	e
-	g -	h -	i	g +	h +	+

= a e i + b f g + c d h - c e g - a f h - b d i

Method B - "co factor expansion" method

---------------------------------------------------------------------------------
ex. evaluate ∆ =

2	3	2
1	4	8
-2	3	6

∆ =

	2	3	2	2	3
	1	4	8	1	4
-	-2 -	3 -	6	-2 +	3  +	+

∆ = (2) (4) (6) + (3) (8) (-2) + (2) (1) (3)

- (2) (4) (-2) - (2) (8) (3) - (3) (1) (6)

∆ = 48 - 48 + 6 +16 - 48 - 18

∆ = -44

"Criss cross" method great for 2x2 and 3x3 determinants.

Problem: It doesn't work for higher-order determinants.

We will, for this reason, learn a general method for calculating determinants called the method of co factors.