Cramers Rule for Matrices You can use properties

Cramer’s Rule for Matrices • You can use properties of matrix determinants for a variety of applications. • Today: – Solving 3 variable systems of equations with Cramer’s rule for determinants – Finding the area of a triangle using Cramer’s rule for determinants

Cramer’s Rule – Systems of Equations • Solve the following system of equations by the old “elimination” method: • x + 4 y –z = 6 • 2 x – y + z = 3 • 3 x +2 y + 3 z = 16

Cramer’s Rule – Systems of Equations • We know from the elimination method that x=1, y=2, and z=3 • Now by Cramer’s Rule: 1. Set up a 3 x 3 matrix using only the coefficients. 2. Find the determinant of the matrix. 3. Replace any column of coefficients with the column of answers. 4. Find the determinant of the slightly altered matrix. 5. Divide the two determinants out for your variable’s value. 6. Repeat process for the other two variables.

Cramer’s Rule – Systems of Equations • Practice this by Cramer’s Rule: • 2 x + 4 y – 3 z = 1 • 3 x – 2 y + 5 z = 8 • x + 7 y – 2 z = -9

Other Applications of the Determinant • Given the Following Coordinate Geometry Triangle: • A (1, 1) • B (2, 6) • C (5, 2) • Find by boxing it in on graph paper and creating a series of right triangles.

Other Applications of the Determinant • Now we’ll solve by using the properties of a determinant. – Set up a matrix – We need an additional row to find a determinant. – Once we have the determinant, let’s use an old Area of a triangle formula