Cramer’s RuleApplying Determinants to

solve Systems of Equations

2x2 & 3x3

2x2 Determinants

Det A = ad – cb

dc

baA det

Cramer’s Rule for 2x2

Part 11. Extract Coefficients

2. Calculate Determinant of Original Matrix

92

1232

yx

yx

21

32

1343122

Part 2 (Solving for x)3. Replace the 1st column of the

coefficient matrix with the constant matrix.

4. Calculate the determinant of new matrix & divide by original determinant.

Cramer’s Rule for 2x2

29

312

21

32

39212 511

51 x

Part 3 (Solving for y)5. Replace the 2nd column of the

coefficient matrix with the constant matrix.

6. Calculate the determinant of new matrix & divide by original determinant.

Cramer’s Rule for 2x2

91

122

21

32

12192 301

30 y

Part 47. To check x and y, substitute 51 in

for x and 30 in for y.

Cramer’s Rule for 2x2

930251

12303512

Part 1: 2 0

1 12

Ex #4 Solve

Part 2: 16 0

6 116 0 16

16

28

x

Part 3: =y

y

2 16

1 64

4

22

3x3 Determinants

cdhbfgaei

Downward

:

hg

ed

ba

ihg

fed

cba

Adet

idbhfag

Upward

ec

:

Cramer’s Rule for 3x3

Part 11. Extract coefficients.

2. Calculate Original Determinant (OD) of Matrix

2 1 2

2 0 1

1 3 4

2 1

2 0

1 3

Cramer’s Rule for 3x3Part 2 (Solving for x)

3. Replace the 1st column of the coefficient matrix with the constant matrix.

4. Calculate the determinant of new matrix & divide by original determinant (15).

7 1 2

5 0 1

10 3 4

7 1

5 0

10 3

40 1 39 x 39

152 6.

Cramer’s Rule for 3x3Part 3 (Solving for y)

5. Replace the 2nd column of the coefficient matrix with the constant matrix.

6. Calculate the determinant of new matrix & divide by original determinant (15).

2 7 2

2 5 1

1 10 4

2 7

2 5

1 10

y 33

152 2.

Cramer’s Rule for 3x3Part 4 (Solving for z)

7. Replace the 3rd column of the coefficient matrix with the constant matrix.

8. Calculate the determinant of new matrix & divide by original determinant (15).

2 1 7

2 0 5

1 3 10

2 1

2 0

1 3

z 3

150 2.

Part 59. To check x and y, substitute 2.6

in for x, 2.2 in for y, and 0.2 in for z.

Cramer’s Rule for 3x3