4.3 Determinants

1
4.3 Determinants Cramers Rule
OBJ To Evaluate determinants of 2 x 2 and 3 x 3
matrices Use Cramer's rule to solve systems of
linear equations

• Determinant of a Matrix A is denoted by
• detA or A
• Determinant of a 2x2
• the difference of the product of the diagonals

ad
- cb
2
Ex Find the determinant of
7(3)
- 2(2)
21 - 4
17
Practice
1(7)
- 2(4)
det

7 - 8
-1
3
Determinant of a 3x3 recopy the 1st two
columns, then subtract the sum of the product of
the diagonals
Ex

• -3
• -2 0
• 1 2

(2)(0)(6)
(-3)(3)(1)
(4)(-2)(2)
(1)(0)(4)
(2)(3)(2)
(6)(-2)(-3)
-
(0 -9 -16) (0 12 36)
-25 - 48
-73
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Cramers Rule
You can use determinants to solve systems of
equations

• Let A be the coefficient matrix
• Linear System Coeff Matrix axbye cxdyf
• If detA 0, then the system has exactly one
  solution

and
5
Ex Solve the system

• 8x5y2
• 2x-4y-10

The coefficient matrix is
and
and
So
(-1, 2)
6
Practice Solve the system

• 2xy1
• 3x-2y-23

The solution is (-3,7)