1. 4.3 Determinants and Cramer’s Rule
Algebra 2

2. Definition
Determinate- A real number associated with any square matrix

3. The Determinate of a Matrix
Determinate of a 2x2 matrix
𝑑𝑒𝑡 𝑎 𝑏 𝑐 𝑑 = 𝑎 𝑏 𝑐 𝑑 =𝑎𝑑−𝑐𝑏
Determinate of a 3x3 matrix
𝑑𝑒𝑡 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 = 𝑎𝑒𝑖+𝑏𝑓𝑔+𝑐𝑑ℎ − 𝑔𝑒𝑐+ℎ𝑓𝑎+𝑖𝑑𝑏

4. Examples Find the determinate of the following matrices 7 2 2 3
−7 0 1 −2 2

5. Example
Find the area of the triangle with coordinates (2, 4), (5, 1), and (2, -2)

6. Area of a Triangle
The area of a triangle with vertices 𝑥 1 , 𝑦 1 , 𝑥 2 , 𝑦 1 , 𝑥 3 , 𝑦 3 is given by
𝐴𝑟𝑒𝑎=± 𝑥 1 𝑦 𝑥 2 𝑦 𝑥 3 𝑦 3 1

7. Example
Find the area of the triangle with coordinates (2, 4), (5, 1), and (2, -2)

8. Example
Find the area of the triangle with vertices (5, -2), (3, 3)

9. Cramer’s Rule
Cramer’s rule: a method (named after a Swiss mathematician Gabriel Cramer) used to solve linear equations
Linear System Coefficient Matrix
𝑎𝑥+𝑏𝑦=𝑒 𝑎 𝑏 𝑐 𝑑
𝑐𝑥+𝑑𝑦=𝑓

10. Cramer’s Rule for a 2x2 matrix
Let A be the coefficient matrix of this linear system
𝑎𝑥+𝑏𝑦=𝑒
𝑐𝑥+𝑑𝑦=𝑓
If det 𝐴≠0, then the system has exactly one solution. The solution is :
𝑥= 𝑒 𝑏 𝑓 𝑑 det 𝐴 and 𝑦= 𝑎 𝑒 𝑐 𝑓 det 𝐴

11. Example Use Cramer’s rule to solve the systems below: 2𝑥+𝑦=1 3𝑥−2𝑦=−23
4𝑥−6𝑦=4
𝑥+5𝑦=14