Download - Chapter Seven Linear Systems and Matrices 7.7 Determinants 7.8 Applications of Determinants
Chapter SevenLinear Systems and Matrices
7.7 Determinants7.8 Applications of Determinants
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 byCramer's rule, a system of linear equations has a unique solution iff thea determinant of the system's matrix is nonzero For example, eliminating x, y, and z from the equations
gives the square coefficient matrix with a unique number which is calledthe determinant for this system of equation. Determinants are defined only for square matrices. If the determinant of a matrix is 0, the matrix is said to be singular, (has no solution)
1 2 3
1 2 3
1 2 3
000
a x a y a zb x b y b zc x c y c z
7.7 – The Determinant of a 2X2 Matrix
The determinant of the matrix is a scalar.
is given by det A = |A| = ad – cb.
a bDet A
c d
Example
Find the determinant of the matrix:
2 93 6
A
15
Example-You Try
Find the determinant of the matrix:
5
2 13 4
A
Example
Find the following determinant using the diagonal method or the minors and cofactors method:
0 2 13 1 24 0 1
14
Example- You Try
Find the following determinant by hand:
34
1 2 04 0 61 3 5
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:
2 0 0 04 2 0 05 6 1 0
1 5 3 3
12
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 = 1 1
2 2
3 3
11 12
1
x yx yx y
Make sure that area is always positive!
1 1 2 2 3 3, , , , ,x y x y x y
Example
• Find the area of the triangle with vertices:
• Draw it
1,0 , 2,2 , 4,3
3 square units2
Example
Find the area of the triangle with vertices:
• Draw it
3,5 , 2,6 , 3, 5
28 square units
Test for collinear points
are collinear if:
1 1
2 2
3 3
11 01
x yx yx y
1 1 2 2 3 3, , , , ,x y x y x y
Example
Determine whether the point are collinear:
0,1 , 4,4 , 8,7yes, they are
Can you think of another way to tell?
Example-You Try
Determine whether the point are collinear:
no, they are not
2, 2 , 1,1 , 7,5
Example
Find x such that the triangle has an area of 4 square units.
4,2 , 3,5 , 1, x
x=3, x=19
Example
Use a determinant to find the equation of a line through (3,5) & (-2,3).
2 195 5
y x
1 1
2 2
11 01
x yx yx y
You Try
Use a determinant to find the equation of a line through (2,1) & (-5,8).
3y x
1 1
2 2
11 01
x yx yx y
Homework 7.7 & 7.8
7.7 page 533 3-11 odd, 31,33, 51
7.8 page 544 1-13 odd, 25