Section 8 7 Determinants and Cramers Rule 2

Section 8. 7 Determinants and Cramer's Rule

2 x 2 Determinant The determinant is a useful value that can be computed from the elements of a square matrix. If the square matrix is a 2 X 2 we cross multiply and find the element. 2 x 2 matrix will be our foundation for larger matrices

Example Find the determinant

3 x 3 determinant If you want to find the determinant of a 3 x 3 matrix you will break it into 3 - 2 x 2 matrices. A row or column in the +/- chart will be used to break the 3 x 3 matrix up The numbers in that specific row or column will also be used as coefficients for the 2 x 2

3 x 3 determinant Given a 3 x 3 matrix

3 x 3 Example

3 x 3 Example Redo the same example with a different row or column

Gabriel Cramer Swiss Mathematician born July 31, 1704 Received his PHD when he was 18 Accomplished most of his work in his 40 s Published Cramer's Rule in 1750

Cramer's Rule The fifth method to solve the system of linear equations. Uses determinants to find the values of intersection. No Solution or Infinite Solutions can not be determined using Cramer's Rule. The linear equations must be in standard form

Steps to solve two unknowns using Cramer's Method Write the linear equations in standard form. Create the first determinant, D, by writing all the coefficients in a 2 x 2 determinant Create the second determinant, Dx, by starting with the D determinant and swapping the x column with the answer column. Create third determinant, Dy, by starting with the D determinant and swapping the y column with the answer column. Define D, Dx, and Dy.

Process

Example Solve the system of linear equations using Cramer's Rule

Example Solve the system of linear equations using Cramer's Rule

Example Solve the system of linear equations using Cramer's Rule

Homework Section 8. 7 # 7, 10, 13, 14, 17, 18, 19