Download Algebra and Trig. I 8.1 – Matrix Solutions to Linear Systems A matrix

Algebra and Trig. I
8.1 – Matrix Solutions to Linear Systems
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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.
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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.
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All the numbers in any row may be multiplied by any nonzero real number.
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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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