Chapter Seven Linear Systems and Matrices 7.7 Determinants 7.8 Applications of Determinants

Precalculus Quiz Review Julie is in charge of ordering the entrees for a school banquet. There are 3 types which cost $22, $38, and $55. She orders 32 entrees and spends $1374. She ordered 8 fewer of the $22 entrees than the $38 entrees. How many of each type of entrée did she order? Write and solve a system of equations.

Ch. 7 Overview Solving Systems of Equations Systems of Linear Equations in Two Variables Multivariable Linear Systems Matrices and Systems of Equations

Ch. 7 Overview (cont.) Operations with Matrices The Inverse of a Square Matrix The Determinant of a Square Matrix Applications of Matrices and Determinants

7.7 – The Determinant of a Square Matrix The Determinant of a 2X2 Matrix The Determinant of a Square Matrix Triangular Matrices

DETERMINANTS Determinants are mathematical objects (scalars) that are very useful in the analysis and solution of systems of linear equations. As shown bysystems of linear equations Cramer's ruleCramer's rule, a system of linear equations has a unique solution iff theiff a determinant of the system's matrix is nonzeromatrixnonzero For example, eliminating x, y, and z from the equations gives the square coefficient matrix with a unique number which is called the determinant for this system of equation. Determinants are defined only for square matrices.square matrices If the determinant of a matrix is 0, the matrix is said to be singular,matrix singular (has no solution)

7.7 – The Determinant of a 2X2 Matrix The determinant of the matrix is a scalar. is given by det A = |A| = ad – cb.

Example Find the determinant of the matrix:

Example-You Try Find the determinant of the matrix:

Example Find the following determinant using the diagonal method or the minors and cofactors method:

Example- You Try Find the following determinant by hand:

7.7 – Triangular Matrices If you have either an upper-triangular, lower- triangular or a diagonal matrix there is a really easy way to find the determinant: Multiply the entries on the main diagonal.

Example Find the following determinant:

7.8 – Applications of Matrices and Determinants Area of a Triangle Test for Collinear Points

Area of a Triangle For a triangle with vertices: Area = Make sure that area is always positive!

Example Find the area of the triangle with vertices: Draw it

Example Find the area of the triangle with vertices: Draw it

Test for collinear points are collinear if:

Example Determine whether the point are collinear: Can you think of another way to tell?

Example-You Try Determine whether the point are collinear:

Example Find x such that the triangle has an area of 4 square units.

Example Use a determinant to find the equation of a line through (3,5) & (-2,3).

You Try Use a determinant to find the equation of a line through (2,1) & (-5,8).

Homework 7.7 & page odd, 31,33, page odd, 25