Cramer’s Rule

Gabriel Cramer was a Swiss mathematician (1704-1752)

Coefficient Matrices You can use determinants to solve a

system of linear equations. You use the coefficient matrix of the

linear system. Linear System Coeff Matrix

ax+by=ecx+dy=f

dc

ba

Cramer’s Rule for 2x2 System Let A be the coefficient matrix Linear System Coeff Matrix

ax+by=ecx+dy=f

If detA 0, then the system has exactly one solution:

A

df

be

xdet

and

A

fc

ea

ydet

dc

ba

Example 1- Cramer’s Rule 2x2 Solve the system: 8x+5y=2 2x-4y=-10

42

58The coefficient matrix is:42)10()32(

42

58

and

So:

42

410

52

xand

42

102

28

y

142

42

42

)50(8

42

410

52

x

242

84

42

480

42

102

28

y

Solution: (-1,2)

Example 2- Cramer’s Rule 2x2

Solve the system: 2x+y=1 3x-2y=-23

The solution is: (-3,7) !!!

Example 3- Cramer’s Rule 3x3 Solve the system: x+3y-z=1 -2x-6y+z=-3 3x+5y-2z=4 1

4

4

253

162

131

453

362

131

z

Let’s solve for Z Z=1

The answer is: (-2,0,1)!!!