MBAD 51415142

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MBAD 5141/5142

• Class 2
• More on Linear Inequalities
• Matrix basics

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Outline for todays class

• Homework help
• Linear Inequalities
• Linear Optimization (Section 11.2)
• Matrix basics (section 9-1)
• Multiplication of Matrices (Section 9.2)
• Solution of Linear Systems by Row Reduction
  (Section 9.3)
• Matrix Inverses (Section 10-1)

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Linear Inequalities

• Linear inequalities are like linear equalities in
  that they consist of two variables (x,y). The
  difference between the two types is that there
  are many possible solutions instead of only one.
  Therefore it is necessary, when graphing, to
  include shaded areas to show solutions.

4
Linear inequality example

• Graph the following linear inequality
• ygt2x - 4

Where do all of the solutions lie? Above or Below
the graph? To the right or left of it?
6
4
2
2
4
6
-2
-4
5
Linear Inequalities continued

• When graphing a system of linear inequalities,
  the intersection of their solution fields is
  important. This type of problem comes up when we
  are interested in a range of values as our
  solution, rather than one unique x or y.

6
Linear Inequalities continued

• Graph the following set of inequalities
• x0, y0, 3x2y 6, x-y 1

• x-y 1

3x2y 6
3
2
X0
1
1
2
3
-1
Y0
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Linear Optimization

• A linear programming problem is one that involves
  finding the maximum or minimum value of some
  linear algebraic expression when the variables in
  this expression are subject to a number of linear
  inequalities.
• Look at example 1 on page 111

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Linear 0ptimization Example
In this type of problem, it is a good idea to
make a table
9
Linear 0ptimization Example (cont.)

• From our problem and table, we can come up with
  the following inequalities
• 1) 5x 3y 105
• 2) 2x 4y 70
• 3) X 0
• 4) Y 0
• We can also determine that our overall weekly
  profit P can be found by using the following
  equation
• P 200x 160y

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Linear 0ptimization Example (cont.)

• We, then, want to find the values of x and y that
  maximize the profit function

y
Maximum Profit Line (slope is same as profit
equation slope). The firm will make maximum
profit when x15 items and y10 items.
X0
5x 3y 105
40
Y0
20
2x 4y 70
0
x
40
20
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Matrices

• In business and other real world scenarios it
  is often the case that more than two variables
  are called for. For example, a firm may produce
  four products A, B, C, and D and each product
  calls for two raw materials A and B. There are,
  then, many combinations (i.e. variables) that can
  be made. Which combination gives the best
  efficiency? Profit? Cost?

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Matrices (continued)

• When there are more than two variables, it is
  more feasible to arrange the variables in an
  array called a matrix. When this is done,
  mathematical operations can be performed on the
  matrices to find solutions. So, since we are
  embarking on a study of matrices, it is important
  to go through a few definitions.

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Matrix definitions

• A matrix (plural matrices) is a rectangular array
  of real numbers, which is enclosed in large
  brackets. Matrices are generally denoted by
  boldface capital letters such as A, B, or C

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Examples of Matrices
The real numbers which form the matrices are
called entries (or elements). The horizontal list
of entries is called the row and the vertical
list of entries is called the column. The size of
a matrix is noted by its number of rows and
columns. Thus, matrix A above is a 2 x 3 matrix
and matrix D is a 1 x 5 matrix. What are the
sizes of B and C?
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More on Matrix definitions

• In general, if A is an m x n matrix, it can be
  written in the following general form

Each element has a unique position (location) in
the matrix.
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More on Matrix definitions

• If all of the elements in a matrix are zero, it
  is called the zero matrix. Any matrix that has
  the same number of rows and columns (i.e., m and
  n are the same) is called a square matrix.

Matrix A is a zero matrix and matrix B is a
square matrix.
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More on Matrix definitions

• Two matrices A and B are said to be equal if they
  are of the same size and their corresponding
  elements are the same. Therefore, the position of
  each element takes on importance. For example

and
Is A B?
Yes, if a2, x5, y0, and b-1and if they are
the same size (yes they are).
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Matrix Operations

• Now that we have been over some definitions, it
  is time to look at some basic operations.
• Matrices can be easily added, subtracted or
  multiplied by a constant (scalar). It is also
  possible to perform combinations of these
  operations.

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Matrix Operations (continued)

• Scalar multiplication is when the matrix is
  multiplied by one constant. In performing this
  operation, multiply the scalar by each element.
  The result is a new matrix of the same size.

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Matrix Operations (continued)

• Example of Scalar Multiplication

then what is 2A?
If
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Matrix Operations (continued)

• When adding or subtracting matrices, perform each
  operation on each individual element. The result
  is a new matrix of the same size.

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Matrix Operations (continued)

• Example of matrix addition/subtraction

Let
and
What is A B?
What is B A?
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Multiplication of Matrices (Section 9-2)

• Finding the product of two matrices requires a
  different method than addition/subtraction of
  matrices or multiplying a matrix by a scalar.
• In short, the elements in individual columns of
  one matrix are multiplied with the elements in
  individual rows of another matrix. The individual
  products are summed and these become the elements
  of the resulting matrix.

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Matrix Multiplication (continued)

• Simple example of matrix multiplication

Let
and
What is AB?
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Matrix Multiplication (continued)

• The previous slide is an example of the basic
  method for finding the product of matrices. Its
  not wise to jump into multiplication without
  knowing some rules, though. The following slides
  give rules, definitions and dos and donts for
  multiplying matrices.

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Matrix Multiplication (continued)

• The product of two matrices can only be found if
  the of columns of one matrix is the same as the
  of rows in another. Furthermore, the order of
  size matters. If finding AB, then the of
  columns in A must be the same as the of rows in
  B. The opposite order will not work. In
  official terms, if A is an m x n matrix and B
  is an n x p matrix then AB will be of size m x p.

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Matrix Multiplication (continued)

• So, it can be concluded that it is possible for
  the product of two matrices to not exist. Try
  finding AB

Let
and
The product does not exist. Why?
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Matrix Multiplication (continued)

• Because the order of size matters in matrix
  multiplication it can also be determined that
  matrix multiplication is not commutative. That is
  AB ? BA.
• Matrix multiplication, however, is associative.
  That is A(BC)(AB)C

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More Matrix Definitions

• A square matrix is called the identity matrix if
  all the elements in its diagonal equal 1. It is
  usually denoted by I and is understood to be the
  same size as the other matrices in the problem.

Size 2 x 2
Size 3 x 3
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More Matrix Definitions

• It is also possible to take a system of linear
  equations and put them in the form of a matrix.
  For example

x

B
A
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Solutions to linear equations using Matrix
methods (Section 9-3)

• Now that we know so much about matrix operations,
  it is time to put that knowledge to use. Since we
  can take linear systems and put them in matrix
  form AXB we can use that fact and begin looking
  at a new method in solving systems. Recall from
  chapter 4 that the solution to a system is a
  unique (x,y) that satisfies both equations. What
  if there are more than two variables and/or more
  than two equations? Matrix algebra gives us a way
  to solve larger systems simultaneously.

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Solutions to linear equations using Matrix
methods (continued)

• A system can be rewritten in augmented matrix
  form

Augmented matrix form is a way to arrange the
coefficients of a system into an array (i.e.,
matrix) without including the variables. We will
see next that once these are arranged into
augmented matrices then by the method of row
reduction we can find the solutions to all
variables, no matter how many there are nor no
matter how many equations there are.
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Row reduction method of solving systems of linear
equations

• Here are the basic rules for row operations. They
  are also found on page 76 of your textbook.
• It is acceptable to interchange any two complete
  rows. It is not acceptable to partially
  interchange.
• It is acceptable to multiply or divide a whole
  row by a nonzero constant. Do not, however do
  this on only parts of a row.
• It is acceptable to add or subtract whole rows to
  or from each other.

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Row reduction method of solving systems of linear
equations

• The object of row reduction is to use the rules
  of row operations to completely reduce an
  augmented matrix into its identity matrix. Once
  this is completed then the resulting column is
  the solution to the system. Look at example 2 on
  page 77-79 of your textbook to get a better feel
  for row reduction. You will find quickly that
  this method is easier with computers!

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Row reduction method of solving systems of linear
equations

• A few helpful hints on row reduction by hand.
• Start with the first column and perform row
  operations consecutively focusing on getting the
  first column completed (i.e., the first element
  is 1, the element below it is 0, the element
  below it is 0, etc.)
• After getting first column set up, work on second
  column in the same manner. Then work on third
  column.
• Use fractions. Decimals will confuse things.
• Write notes beside each matrix to keep track of
  what row operation you are performing.
• See box on page 80 for recap of these rules

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Inverses of matrices

• This stuff only applies to square matrices.
• If ABI and BAI, then B is said to be an inverse
  of A.
• Not every square matrix has an inverse.

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Inverses of matrices (continued)

• The inverse of a matrix (noted as A¹) can be
  found by setting up an augmented matrix with its
  inverse. Just row reduce the A part and the
  augmented matrix will become the inverse.

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Inverse Matrix Homework (in addition to what else
you are assigned)

• Page 100-101 3,7,9