Matrices: Basic Concept

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Matrices Basic Concept
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Matrices Introduction
A matrix, in general sense, represents a
collection of information stored or arranged in
an orderly fashion. The mathematical concept of a
matrix refers to a set of numbers, variables or
functions ordered in rows and columns. Such a set
then can be defined as a distinct entity, the
matrix, and it can be manipulated as a whole
according to some basic mathematical rules.
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Matrices Introduction

• A matrix is a set of elements, organized into
  rows and columns

rows
columns
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Example 1
A matrix with 9 elements is shown below.
Matrix A has 3 rows and 3 columns. Each
element of matrix A can be referred to by its
row and column number. For example,
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Example 2
A computer monitor with 800 horizontal pixels and
600 vertical pixels can be viewed as a matrix of
600 rows and 800 columns.
In order to create an image, each pixel is filled
with an appropriate color.
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Matrices Basic Concept

• Order of a Matrix
• The order of a matrix is defined in terms of its
  number of rows and columns.
• Column Matrix
• A matrix with only one column is called a column
  matrix or column vector.

Order of a matrix No. of rows No. of columns
Matrix A, therefore, is a matrix of order 33.
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Matrices Basic Concept

• Row Matrix
• A matrix with only one row is called a row
  matrix or row vector.
• Square Matrix
• A matrix having the same number of rows and
  columns is called a square matrix.

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Matrices Basic Concept

• Rectangular Matrix
• A matrix having unequal number of rows and
  columns is called a rectangular matrix.
• Real Matrix
• A matrix with all real elements is called a real
  matrix.

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Trace of A Matrix

• Principle Diagonal and Trace of a Matrix
• In a square matrix, the diagonal containing the
  elements a11, a22, a33, a44, , ann is called
  the principal or main diagonal.
• The sum of all elements in the principal
  diagonal is called the trace of the matrix.

The trace of the matrix is 2 3 9 14.
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Unit and Zero Matrices

• Unit Matrix
• A square matrix in which all elements of the
  principal diagonal are equal to 1 while all other
  elements are zero is called the unit matrix.
• Zero or Null Matrix
• A matrix whose elements are all equal to zero is
  called the null or zero matrix.

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Diagonal Matrix

• If all elements except the elements of the
  principal diagonal of a square matrix are zero,
  the matrix is called a diagonal matrix.

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Transpose of a Matrix
The transpose AT of an mn matrix A is the
nm matrix obtained by interchanging the rows and
columns of A.
Transpose of a sum Transpose of a
product (AB)T AT BT (A B)T
BTAT
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Matrices Basic Concept
A matrix A is said to be symmetric if aij aji
for all i and j. A AT Example
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Matrices Basic Concept

• Row Rank of a Matrix
• The maximum number of linearly independent rows
  of a matrix A is called the rank of A.

Rank A
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Matrices Basic Concept

• Augmented Matrix
• A matrix made up of both the L.H.S. and R.H.S.
  of a system of linear equations. It is denoted by

See following for examples. The augmented matrix
will become quite important in the near future.
In the following system of linear equations all
three equations are linearly independent.
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Matrices Basic Concept
Therefore, the rank of A will be 3. The
augmented matrix for the system is
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Matrices Basic Concept
Consider the following linear systems with 2
independent equations.
In the above set, Eqn. (3) can be generated by
adding Eqn. (1) to Eqn. (2). Therefore, Eqn. (3)
is a dependent equation.
Therefore, the rank of A will be ? The
augmented matrix for the system is
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Basic Operations

• Addition, Subtraction, Multiplication

Just add elements
Just subtract elements
Multiply each row by each column
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Multiplication

• Is AB BA? Maybe, but maybe not!
• Heads up multiplication is NOT commutative!

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Properties of Matrix Multiplication

• NOT Commutative
• A B ?B A

• Associative
• A(BC) (AB)C
• A m by n
• B . n by p
• C . p by r
• Product Order?

• Distributive
• A(BC) AB AC
• A m by r
• C and B must be r by n
• Product Order m by n

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Numerical example of the product rule
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Equality of Matrices
Two matrices are equal if all corresponding
elements are equal.
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Multiplication by a Scaler
If a matrix is multiplied by a scalar k, each
element of the matrix is multiplied by k.
Example
If Decembers production is doubled in January,
what will be the unit production in January?
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Example of Matrix Addition

• A manufacturing plant produces 3 models of
  widgets A, B, and C. In November they produced
  23, 16 and 10 units A, B, and C. In December they
  produced 18, 12, and 9 units of A, B, and C.
• What was the total product for each unit for
  these two months?

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Example of Matrix Multiplication

• Consider the following problem To manufacture a
  widget requires 23 units of material and 7 units
  of labor. The cost per unit of material is 450,
  and per unit of labor is 600.
• What is the total cost of producing the widget?

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Example of Matrix Multiplication

• One months widget production is as follows
• Model A B C
• Material 10 5 2
• Labor 6 4 4
• Costs
• Material is 15 per unit
• Labor is 6 per unit
• What is total cost to manufacture 3 models?

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Examples

• Find the value of m such that the following
  equation is correct
• Evaluate A2-5A4I if