Download Chapter 8: Matrices and Determinants

Chapter 8: Matrices and Determinants
Tuesday June 1: 8-1: Matrix Solutions to Linear Systems. Gauss-Jordan Elimination.
After today’s lesson you should be able to do the following:
1. Write the augmented matrix for a linear system
2. Perform matrix row operations
3. Use matrices and Guass-Jordan elimination to solve systems
Objective 1: Writing an augmented matrix given a system of equations
A matrix gives us a shortened way of writing a system of equations. The first step in solving a system of linear equations
using matrices is to write the augmented matrix. An augmented matrix has a vertical bar separating the columns of the
matrix into tow groups. The coefficients of each variable are placed to the left of the vertical line and the constants are
placed to the right. If any variable is missing, its coefficient is 0:
In 1-2, Write the augment matrix that represents the given system of equations.
x  3y  9
x  2 y  3z  9
1.  y  4 z  2
2.
 x  3 y  4
2 x  5 y  5 z  17
x  5z  0
Objective 2: Perform matrix row operations
Elementary Row Operations
1. Interchange two rows
2. Multiply a row by a nonzero constant
3. Add a multiple of a row to another row.
Examples of Row Operations
(a) Interchange the first and second Rows.
Original Matrix
New Row-Equivalent Matrix
 0 1 3 4


 1 2 0 3 
 2 3 4 1 
(b) Multiply the first row by
Original Matrix
 2 4 6 2 


 1 3 3 0 
 5 2 1 2 
1
.
2
New Row-Equivalent Matrix
(c) Add −2 times the first row to the third row.
Original Matrix
New Row-Equivalent Matrix
1 2 4 3 


 0 3 2 1
 2 1 5 2 
Math Analysis Notes Prepared by Mr. Hopp
1
Objective 3: Use Gauss-Jordan Elimination to solve system of equations.
With Guass-Jordan elimination, elementary row operations are applied to a matrix to obtain a row-equivalent matrix that is in a form
called reduced row-echelon form. For a system of three equations a matrix in reduced row-echelon form is:
1 0 0 a 


0 1 0 b 
0 0 1 c 
Notice that reduced row-echelon form contains a diagonal of ones with zeros in all other positions of the matrix. When you convert a
matrix that is in reduced row-echelon form to a system of equations you have the solutions:
1 0 0 a 
xa


0 1 0 b  means y  b
0 0 1 c 
zc
x  2 y  3x  9
In 3, Use Guass-Jordan elimination to solve the system:
 x  3 y  4
2 x  5 y  5 z  17
Math Analysis Notes Prepared by Mr. Hopp
2
Wed June 2: 8-2 Inconsistent and Dependent Systems and Their Applications
After today’s lesson you should be able to do the following:
1. Apply Gaussian elimination to systems without unique solutions.
2. Apply Gaussian elimination to systems with more variables than equations.
3. Solve problems involving systems without unique solutions.
Objective 1: Apply Gaussian elimination to systems without unique solutions.
x  y  2z  4
In 1, Use Gaussian elimination to solve the system:
xz 6
2x  3 y  5z  4
3x  2 y  z  1
x  2y  z  5
In 2, Use Gaussian elimination to solve the system: 2 x  5 y  3z  6
x  3y  4z  1
Math Analysis Notes Prepared by Mr. Hopp
3
Objective 2: Apply Gaussian elimination to systems with more variables than equations.
When we have more variables than equations the process is similar to example #2 above when we get a true statement of 0 = 0. You
need to define your solution in terms of z.
In 3, Use Gaussian elimination to solve the system:
x  2 y  3z  70
x  y  z  60
Objective 3: Solve problems involving systems without unique solutions.
In a network, which can be a computer circuit or the intersections of streets, it is assumed that the total flow into a junction is equal to
the total flow out of the junction. For example:
x
20
1
This sample network #1 can be represented
by the following equation:
y
Math Analysis Notes Prepared by Mr. Hopp
4
In 4, The flow of traffic (in vehicles per hour) through a network of streets is shown in the figure:
400
300
(a) Set up a system of linear equations involving x, y, and
x
1
2
350
200
z.
z
y
3
(b) Put the systems of equations found in part (a) into
matrix form and solve the system using Gaussian
elimination.
(c) If construction limits the value z to 400 cars per hour,
how many cars per hour must pass between the other
intersection to keep traffic flowing?
450
700
Thur June 3 Quiz on Gaussian Elimination to find the solution to a system of equations
Math Analysis Notes Prepared by Mr. Hopp
5
Friday June 4: 8-3 Matrix Operations and Their Applications
After today’s lesson you should be able to do the following:
1. Use matrix notation
2. Understand what is meant by equal matrices
3. Add and subtract matrices
4. Perform scalar multiplication
5. Multiply matrices
Objective 1: Use matrix notation
We have seen that an array of numbers arranged in rows and columns and placed in brackets is called a matrix. We can represent a
matrix in two different ways.
 A capital letter, such as A, B, or C, can denote a matrix.
 A lower case letter enclosed in brackets: A   aij  can also denote a matrix.
A matrix of order m×n has m rows and n columns. If m = n the matrix is called a square matix.
Practice: (a) State the order of the matrix and (b) identify a12 and a31 .
 5 2
A   3  
 1 6 
Objective 2: Understand what is meant by equal matrices.
Two matrices A and B are equal if and only if they have the same order m×n and each entry in the matrix are the same.
 x
Practice: Find the value of x, y, z given A = B: A  
z  2
y  1
 9 12 
and B  


6 
15 6 
Objective 3: Add and subtract matrices.
Matrices of the same order can be added or subtracted by adding or subtracting the elements in corresponding positions. Matrices that
are not the same order can not be added or subtracted.
 5 4
 4 8 
Practice: Given A   3 7  and  6 0 
 0 1 
 5 3 
Find (a) A + B
(b) A – B
(c) B – A
Math Analysis Notes Prepared by Mr. Hopp
6
Objective 4: Perform scalar multiplication.
To multiply a matrix A by a scalar c we multiply each entry in A by c.
 w x  cw cx 
c


 y z   cy cz 
 4 1 
 1 2 
Practice: If A  
and B  

 , find the following matrices:
 3 0
8 5
(a) −6B
(b) 3A + 2B
 2 8
 10 1 
Practice: Solve for X in the matrix equation 3X + A = B, where A  
and B  

.
0 4 
 9 17 
Objective 5: Multiply matrices.
In order to multiply matrices you must think of it as row-by-column multiplication:
a b   e
c d   g


f   ae  bg

h   ce  dg
af  bh 
cf  dh 
 1 3
4 6
Practice: Find AB given A  
and B  


 2 5
1 0 
Math Analysis Notes Prepared by Mr. Hopp
7
In order to multiply matrices AB the number of columns of A must equal the number of rows of B. The product of AB will have an
order equal to the number of columns of A × number of rows of B.
Matrix A
Matrix B
m×n
n×p
These must
be equal
The order of AB
is m×p
Practice: Where possible, find each product
1 3   2 3 1 6 
(a) 


0 2  0 5 4 1 
 2 3 1 6  1 3 
(b) 


 0 5 4 1  0 2 
Math Analysis Notes Prepared by Mr. Hopp
8
Mon June 7: 8-4 Multiplicative Inverses of Matrices and Matrix Equations
After today’s lesson you should be able to do the following:
1. Find the multiplicative inverse of a square matrix
2. Use inverse to solve matrix equations
Objective 1: Find the multiplicative inverse of a square matrix
1
1
 1 and  n  1 .
n
n
We define the multiplicative inverse of a square matrix in a similar manner. However, the matrix that we use for “1” has 1s diagonal
from upper let to lower right and 0s elsewhere. We call a square matrix with 1s on the diagonal and zeros elsewhere the identity
matrix:
1 0 0 0 
1 0 0 
0 1 0 0 
1 0 




I2  
 I 3  0 1 0  I 4  0 0 1 0 
0 1 
0 0 1 


0 0 0 1 
Definition of the Multiplicative Inverse of a Square Matrix
Let A be an n×n matrix. If there exists an n×n matrix A−1 (read “A inverse”) such that AA−1 = I and A−1A = I.
then A−1 is the multiplicative inverse of A.
Remember that the multiplicative inverse of a number n has the following property: n 
To show that a matrix B is the multiplicative inverse of A you must show both AB = I and BA = I.
 2 1
 1 1
Practice: Show that B is the multiplicative inverse of A, where A  
and B  

.
 1 1
 1 2 
To find the Multiplicative Inverse of a 2×2 Matrix
a b 
1  d b 
If A  
, then A1 
.

ad  bc  c a 
c d 
The matrix A has an inverse if and only if ad – bc ≠ 0. If ad – bc = 0, then A does not have a multiplicative inverse.
1 2 
Practice: Find the inverse of the matrix A  

3 4 
Math Analysis Notes Prepared by Mr. Hopp
9
The above method only works for 2×2 matrices. For other square matrices you must use the following procedure.
Procedure for Finding the Multiplicative Inverse of an Invertible Matrix:
1. Form the augmented matrix  A I n  , where In is the multiplicative identity matrix of the same order as the given matrix A.
2. Perform row operations on  A I n  to obtain a matrix of the form  I n B  .
3. Matrix B is A−1
4. Verify the result by showing that AB = I and BA = I
 1 0 2
Practice: Find the multiplicative inverse of : A   1 2 3


 1 1 0 
Math Analysis Notes Prepared by Mr. Hopp
10
Solving a System Using A−1
If AX = B has a unique solution, X = A−1B. To solve a linear system of equations multiply A−1 and B to find X.
x  2z  6


Practice: Solve the system using A , the inverse of the coefficient matrix:  x  2 y  3z  5

x y 6

−1
Math Analysis Notes Prepared by Mr. Hopp
11
Tuesday June 8: 8-5 Determinants and Cramer’s Rule
After today’s lesson you should be able to do the following:
1. Evaluate a second-order determinant
2. Solve a system of linear equations in two variables using Cramer’s rule
3. Evaluate a third-order determinant
4. Solve a system of linear equations in three variables using Cramer’s rule
Objective 1: Evaluate a determinant of a 2×2 matrix
Definition of the Determinant of a 2×2 Matrix
a b
a b 
The determinant of the matrix 
is denoted by
and is defined by:

c d
c d 
a b
 ad  cb .
c d
Practice: Evaluate the determinant of each of the following matrices:
10 9 
4 3
(a) 
(b) 


 6 5
 5 8
Objective 2: Solve a system of linear equations in two variables using Cramer’s rule.
Solving a Linear System in Two Variable Using Determinants aka Cramer’s Rule:
c
ax  by  c
b
a
c
f e
d f
and y 
a
b
a b
dx  ey  f
d e
d e
Notice that in order to solve for x you replace the coefficients of x with the answer coefficients. When you are solving for y you
replace the coefficients of y with the answer coefficients.
If
then x 
 5 x  4 y  12
Practice: Use Cramer’s rule to solve the system: 
3 x  6 y  24
Objective 3: Evaluate a determinant of a 3×3 Matrix.
Math Analysis Notes Prepared by Mr. Hopp
12
To find the determinant of a 3×3 Matrix use the following pattern:
a b
c
d
f a
e
g h
i
e
f
h
i
d
b c
h i
g
b
c
e
f
 2 1 7
Practice: Evaluate the determinant of:  5 6 0  .


 4 3 1 
Objective 4: Solve a system of linear equations in three variables using Cramer’s Rule.
Step 1: Find the determinant of the coefficient matrix, called D.
Step 2: Find the determinant with the 1st column of the coefficient matrix replaced with the solutions to each linear equation, called
Dx.
Step 3: Find the determinant with the 2nd column of the coefficient matrix replaced with the solutions of each linear equation, called
Dy.
Step 4: Find the determinant with the 3rd column of the coefficient matrix replaced with the solutions of each linear equation, called
D z.
Dy
D
D
, and z  z .
Step 5: Solve for x, y, and z by using the following: x  x , y 
D
D
D
 3x  2 y  z  16

Practice: Use Cramer’s rule to solve the system: 2 x  3 y  z  9
 x  4 y  3z  2

Math Analysis Notes Prepared by Mr. Hopp
13