Download Pre-Calculus. I 8.1 – Matrix Solutions to Linear Systems A matrix is a

Pre-Calculus. I
8.1 – Matrix Solutions to Linear Systems
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
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A matrix is a rectangular array of elements.
o An array is a systematic arrangement of numbers or symbols in rows and
columns.
Matrices (the plural of matrix) may be used to display information and to solve systems
of linear equations.
The numbers in the rows and columns of a matrix are called the elements of the matrix.
Matrices are rectangular arrays of numbers that can aid us by eliminating the need to
write the variables at each step of the reduction.
For example, the system
2 x  4 y  6 z  22
3x  8 y  5z  27
 x  y  2z  2
may be represented by the augmented matrix
Dimensions of a Matrix
 The dimensions of a matrix may be indicated with the notation r  s, where r is the
number of rows and s is the number of columns of a matrix.
 A matrix that contains the same number of rows and columns is called a square matrix.
Example: 3  3 square matrices:
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Example -Write the augmented matrix of the given system of equations.
3x+4y=7
4x-2y=5
Example -Write the system of equations corresponding to this
augmented matrix. Then perform the row operation
on the given augmented matrix. R 2  4r1  r2
 1 3 3 5


 4 5 3 5
 3 2 4 6 
Augmented Matrix
 The first step in solving a system of equations using matrices is to represent the system of
equations with an augmented matrix.
o An augmented matrix consists of two smaller matrices, one for the coefficients of the
variables and one for the constant
Systems of equations Augmented Matrix
a1x + b1y = c1
a2x + b2y = c2
 a1 b1 c1 

 a2
b2

c2 
Row Transformations
 To solve a system of equations by using matrices, we use row transformations to obtain
new matrices that have the same solution as the original system.

We use row transformations to obtain an augmented matrix whose numbers to the left of
the vertical bar are the same as the multiplicative identity matrix.
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Procedures for Row Transformations
 Any two rows of a matrix may be interchanged.

All the numbers in any row may be multiplied by any nonzero real number.

All the numbers in any row may be multiplied by any nonzero real number, and these
products may be added to the corresponding numbers in any other row of numbers.
A matrix with 1’s down the main diagonal and 0’s below the 1’s is said to be in row echelon
form (ref). We use row operations on the augmented matrix. These row operations are just like
what you did when solving system of equations using addition method, this method is called
Gaussian Elimination.
A matrix with 1’s down the main diagonal and 0’s above and below the 1’s is said to be in
reduced row echelon form (rref). We use row operations on the augmented matrix. These row
operations are just like what you did when solving system of equations using addition method.
To Change an Augmented Matrix to the Reduced Row Echelon Form (rref)
Use row transformations to:
1.
Change the element in the first column, first row to a 1.
2.
Change the element in the first column, second row to a 0.
3.
Change the element in the second column, second row to a 1.
4.
Change the element in the second column, first row to a 0.
Sometimes it is advantageous to write a matrix in reduced row echelon form. In this form, row
operations are used to obtain entries that are 0 above as below the leasing 1 in a row. The
advantage is that the solution is readily found without needing to back substitute.
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Graphing Calculator-Matrices, Reduced Row Echelon Form
To work with matrices we need to press 2nd x-1 key to get to the matrix menu.
To type in a matrix, cursor to the right twice to get to EDIT, press ENTER and type in the
dimensions of matrix A. Press ENTER after each number, then type in the numbers that
comprise the matrix, again pressing ENTER after each number.
To get out of the matrix menu press QUIT (2nd MODE)
To get to reduced row echelon form bring up the matrix menu (2nd x-1) and cursor to MATH.
Cursor down to B, rref (reduced row echelon form). Press ENTER
To type the name of the matrix you want to work with, again bring up the matrix menu (2nd x-1)
and under names, choose the letter of the matrix you are working with, press ENTER again.
Example: Solve this system of equations using matrices (row operations).
3x  5 y  3
15 x  5 y  21
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Example: Solve the system of equations using matrices. If the system has no solution, say
inconsistent.
Example:
Write the system of equations corresponding to this
augmented matrix. Then perform the row operation
on the given augmented matrix. R 2  4r1  r2
 1 3 3 5


 4 5 3 5
 3 2 4 6 
A System of Equations with an Infinite Number of Solutions
Example: Solve the system of equations given by
x  2 y  3 z  2
3x  y  2 z  1
2 x  3 y  5z  3
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A System of Equations That Has No Solution
Example: Solve the system of equations given by
x y z 1
3x  y  z  4
x  5 y  5 z  1
Systems with no Solution
If there is a row in the augmented matrix containing all zeros to the left of the vertical line and a
nonzero entry to the right of the line, then the system of equations has no solution.
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8.3 – Matrix Operations and Their Applications
Notations for Matrices
We can represent a matrix in two different ways.
1. A capital letter, such as A, B, or C, can denote a matrix.
2. A lowercase letter enclosed in brackets, such as that
shown below can denote a matrix.
A=  ai j 
A general element in matrix A is denoted by a i j . This
refers to the element in the i th row and jth column.
a32 is the element located in the 3rd row, 2nd column.
See below.
 a11

 a21
a
 31
a12
a22
a32
a13 

a23 
a33 
A matrix of order m  n has m rows and
n columns. If m=n, a matrix has the same
number of rows as columns and is called
a square matrix.
Example -
 1 3 3 
Let A=  4 5 3
 3 2 4 
a. What is the order of A?
b. If A=  ai j  , identify a 23 , and a 31
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Equality of Matrices
Matrix Addition and Subtraction
0 0
Zero matrix = 

0 0
 3
Additive Inverses: The inverse of 
1
 3 2   3


 1 5   1
2  3
 is 
5   1
2   0

5   0
2 

5 
0

0
 3 2 
 3 2 
If A= 
 then -A = 

 1 5
 1 5 
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Example -
Example -
Subtract the following two matices.
 3 1  2 1 



 0 4   4 3 
What is the zero matrix for all 2  3 matrices?
What is the additive inverse for the matrix below?
 3 1


0 4 
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Scalar Multiplication
Example -
 2 3 
 6 3 
If A= 
 and B= 

 5 4 
 1 0 
what is 2A+3B
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Matrix Multiplication
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Notice that when multiplying two matrices the order of the
product matrix is the order of the two outside dimensions
from the original matices. Because matrix multiplication
is not commutative be careful about the order in which
matrices appear when
performing this operation.
4
AB = 1 2 3 5   32
6 
4
 4 8 12


BA  5  1 2 3  5 10 15
6 
6 12 18
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Find the product.
Example -
 4 
 2 1 3  1 
 2 
Find the product.
Example -
 4 
 1  2 1 3

 
 2 
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Find the product
Example -
 1
0

3

1
2
1   4 0 
1  2 3

2
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8.5 – Determinants and Cramer’s Rule
The Determinant of a 2 x 2 Matrix
Example : Evaluate the determinant of each of the following matices:
 2 3
a. 

 5 1
 3 2 
b. 

 4 1
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How to Calculator:
To evaluate determinates in your calculator. Press 2nd x-1 then scroll over to EDIT to input your
matrix. Then 2nd MODE to get out of the matrix menu. Then go back into the matrix menu 2nd
x-1 scroll over to MATH and down to det (determinate). Press ENTER and then go back into
matrix menu to call the matrix that you want ot take the determinate of.
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Example : Evaluate the determinant of the following matrix:
 2 1 0 
 1 1 2 


 3 1 0 
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The Determinant of Any N x N Matrix
The determinant of a matrix with n rows and n columns is said
to be an nth-order determinant. The value of an nth-order determinant
can be found in terms of determinants of order n-1.
We can generalize the idea for fourth-order determinants and
higher. We have seen that the minor of the element a i j is the
determinant obtained by deleting the ith row and the jth column
in the given array of numbers. The cofactor of the element a i j
is (-1)i  j times the minor of the a ij th entry. If the sum of the row
and column (i+j) is even, the cofactor is the same as the minor.
If the sum of the row and column (i+j) is odd, the cofactor is the
opposite of the minor.
Example: Find the determinate of the following matrix.
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