3.5 Determinants and
Cramer’s Rule
The Determinant of a Matrix


The determinant of a matrix A is
denoted by |A|.
Determinants exist only for square
matrices.
The Determinant for a 2x2 matrix

If A =

Then

This one is easy
a b 
c d 


A  ad  bc
Cramer’s Rule



Let D be the coefficient matrix
Linear System
ax+by = e
cx+dy = f
Coeff Matrix
a b 
c d 


If det D ≠ 0, then the system has exactly one solution:
e b
Dx
f d
x

a b
D
c d
and
a e
Dy
c f
y

a b
D
c d
Example 1- Cramer’s Rule (2x2)

Solve the system:
8x + 5y = 2
2x ─ 4y = −10
The coefficient matrix is:
So:
2
5
 10  4
x
 42
8 5 
 2  4


and
and
8
5
2 4
 (32)  (10)  42
8 2
2  10
y
 42
Example 1 (continued)
2
5
 10  4  8  (50)
42
x


 1
 42
 42
 42
8 2
2  10  80  4  84
y


2
 42
 42
 42
Solution: (-1,2)
The Determinant for a 3x3 matrix

a1 b1
Value of 3 x 3 (4 x 4, 5 x 5, etc.)
determinants can be found using
so called expansion by minors.
c1
a 2 b 2 c 2  a1
a 3 b 3 c3
b2 c2
b 3 c3
 b1
a 2 c2
a 3 c3
 c1
a 2 b2
a 3 b3
Example 2 - Cramer’s Rule (3x3)

Solve the system:
x + 3y – z = 1
–2x – 6y + z = –3
3x + 5y – 2z = 4 z 
1
3
2 6 3
3
1
5
3
2 6
Let’s solve for Z
The answer is: (2,0,1)!!!
1
3
5
4
4

1
1  4
1
2