4.3 Determinants & Cramer’s Rule

Objectives/Assignment

Warm-UpSolve the system of equations:

What is the product of these matrices?

4 8

5 6

(2,1)

Associated with each square matrix is a real number called

it’s determinant.

We write The Determinant of matrix A as det A or |A|

Here’s how to find the determinant of a square 2 x 2 matrix:

4 8

5 6

Multiply

Multiply24 (1st)

40 (2nd )

Now subtract these two numbers.

- = -16-16 is the determinant of this matrix

24 (1st) 40 (2nd )

In General

det ad bca b ac

c d b d

Determinant of a 3 x 3 Matrix

det

a b c

d e f

g h i

a b c

d e f

g h i

a b

d e

g h

(aei+bfg +cdh)

(gec +hfa +idb)Now Subtract the 2nd set products from the 1st.

(aei + bfg + cdh) - (gec + hfa + idb)

Compute the Determinant of this 3 x 3 Matrix

2 1 3

det 2 0 1

1 2 4

2 1 3

2 0 1

1 2 4

2 -1

-2 0

1 2

(0+ -1 -12)

(0 +4 +8)Now Subtract the 2nd set products from the 1st.

(-13) - (12) =-25

You can use a determinant to find the Area of a Triangle

(a,b)

(c,d)

(e,f)

The Area of a triangle with verticies (a,b), (c,d) and (e,f) is given by:

11

12

1

a b

c d

e f

Where the plus or minus sign indicates that the appropriate sign should be chosen to give a positive value answer for the Area.

In general the solution to the systemax + by = e

cx + dy = fis (x,y)

where

x=

y=

a b

c d

a b

c d

e b

f d

a e

c f

anda b

c d= 0

You can use determinants to solve a system of equations. The method is called Cramer’ rule and named after the Swiss mathematician Gabriel Cramer (1704-1752). The method uses the coefficients of the linear system in a clever way.

If we let A be the coefficient matrix of the linear system, notice this is just det A.

Use Cramer’s Rule to solve this system:

ax + by = e

cx + dy = f

x= a b

c d

e b

f d

y= a b

c d

a e

c f

x= 4 2

5 1

10 2

17 1

y= 4 2

5 1

4 10

5 17

=

=

(10)(1) –(17)(2)

(4)(1) –(5)(2)

(4)(1) –(5)(2)

(4)(17) –(5)(10)

=

=68 - 50

10 - 34

4 - 10

4 - 10=

18 -6

= -3

The system has a unique solution at (4,-3)

=-24 -6

= 4

4x + 2y = 105x + y = 171

Solve the following system of equations using Cramer’s Rule:

ax + by = e

cx + dy = f

x= a b

c d

e b

f dx=

6 4

3 2

10 4

5 2=

(10)(2) –(5)(4)

(6)(2) –(3)(4)=

20 - 20

12 - 12=

0 0

6x + 4y = 103x + 2y = 5

Since, the determinant from the denominator is zero, and division by zero is not defined: THIS SYSTEM DOES NOT HAVE A UNIQUE SOLUTION and Cramer’s Rule can’t be used.

Cramer’ Rule can be use to solve a 3 x 3 system.

det

j

k

a b

b e

c h lz

A

Let A be the coefficient matrix of this linear system:

ax by cz

dx ey fz

gx hy

j

lz

k

i

a b c

A d e f

g h i

If det A is not 0, then the system has exactly one solution. The solution is:

det

b c

e fk

l h i

j

xA

det

a c

d fk

g l i

j

yA