4.3 Determinants and Cramer’s Rule Algebra 2

Definition Determinate- A real number associated with any square matrix

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

Examples Find the determinate of the following matrices 7 2 2 3 4 3 1 5 −7 0 1 −2 2

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

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=± 1 2 𝑥 1 𝑦 1 1 𝑥 2 𝑦 2 1 𝑥 3 𝑦 3 1

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

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

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

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

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