# Aim: Determinants and Cramer's law ```Aim: Determinants and Cramer’s law
Do Now:
1. Solve for x and y in terms of a,b,c,d,e,f
Use elimination to eliminate y and solve for x and vice versa
ax + by = c
dx + ey = f
(ce - bf)
x=
(ae - bd)
2. Solve for x and y in terms of a,b,c,d,e,f,g,h,i,j,k,l
ax + by +cz = d
ex + fy + gz = h
ix + jy + kz = l
Determinant of:
=a bG
c d
=3 7 G
= (3 # 10) - (7 # 5) = - 5
5 10
To find the determinant of a matrix with a calculator:
Matrix  type in matrix  quit  matrix  math  det  insert the matrix 
enter
Cramer’s Rule:
3x – 2y = 4
6x + y = 13
=3 - 2 G
= (3 # 1) - (- 2 # 6) = 15
D=
6 1
Dx = replace column x with the answers to the equations:
=4 - 2 G
= (4 # 13) - (- 2 # 13) = 30
Dx =
13 1
Dy = replace column y with the answers to the equations:
=3 4 G
= (3 # 13) - (4 # 6) = 15
Dy =
6 13
Cramer’s law states that the solutions of
Dx
D
Dy
=
y
D
x=
So in this case:
30 =
2
15
15 =
y=
1
15
x=
Cramer’s Rule for 3 by 3 Systems of Equations:
3x + y – z = 2
-x +2y + 4z = 15
x – 2y -3z = -12
To solve a 3 by 3 determinant you do something called expand along the first row.
Take the first number and then times it by the determinant of the 2 by 2 matrix left
after the row and column the first number is in is crossed out. Also remember to
alternate signs.
&gt;3
D =
- 1
2
&gt;
=+ 3#
2
1
- 1
15
2
4
- 12
- 2
- 3
3
2
- 1
- 1
15
4
1
- 12
- 3
&gt;
Dy =
4
H
=2
- 2 - 3
1
Dx =
- 1
1
4
- 2 - 3
H
= 7
H
= 14
G =- 1 4 G =- 1
-
+- 1
1
3
1
2
- 2
G
= 7
&gt;
Dz =
3
1
2
- 1
2
15
1
- 2
- 12
H
= 21
After applying Cramer’s rule we get
Dx
=
D
Dy
=
=
y
D
Dz
=
z=
D
x=
Doris Xia
7 =
1
7
14 =
2
7
21 =
3
7
```