1. 4.3 Determinants & Cramer’s Rule
OBJ: To Evaluate determinants of 2 x 2 and 3 x 3 matrices & Use Cramer's rule to solve systems of linear equations
4.3 Determinants & Cramer’s Rule
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) =
= 17
Practice:
=1(7)
- 2(4)
det =
= 7 - 8
= -1

3. Determinant of a 3x3:. recopy the 1st two columns, then subtract the
Determinant of a 3x3: recopy the 1st two columns, then subtract the sum of the product of the diagonals
Ex: -3 =
(2)(0)(6)
+ (-3)(3)(1)
+ (4)(-2)(2) -
(1)(0)(4)
+ (2)(3)(2)
+ (6)(-2)(-3)
= ( ) – ( ) =
= -73

4. Cramer’s Rule Let A be the coefficient matrix
****You can use determinants to solve systems of equations:
Cramer’s Rule
Let A be the coefficient matrix
Linear System Coeff Matrix ax+by=e cx+dy=f
If detA 0, then the system has exactly one solution:
and

5. Ex: Solve the system (-1, 2) 8x+5y=2 2x-4y=-10
The coefficient matrix is:
and
and
So:
(-1, 2)

6. Practice: Solve the system
2x+y=1
3x-2y=-23
The solution is: (-3,7)