matrix algebra
DESCRIPTION
TRANSCRIPT
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An introduction to Matrix Algebra
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Algebra
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MATRIXA matrix is an ordered rectangular array of numbers, arranged in rows and columns.
columns
rows
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ORDER OF A MATRIX
The size or order of a matrix is described by its number of rows and the number of columns.
If a matrix, A, has m rows and n columns then A is described as an mxn matrix.
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The numbers in a matrix are called its elements. The element in the ith row and jth column of a matrix is generally denoted by aij. A matrix with m rows and n columns is written or .
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Row Matrix
A matrix with just one row is called a row matrix (or row vector).
jn aaaaA , 2 1 (1 x n)
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Column Matrix
i
m
a
a
a
a
A 2
1
A matrix with just one column is called a column matrix.
(m x 1)
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Matrices of the same order
Two matrices which have the Same number of rows and columns are said to be matrices of the same order.
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Equal MatricesTwo matrices A = (aij) and B = (bij) are said to be equal if, and only if, each element aij of A is equal to the corresponding element bij of B.
In symbolic form this reads:
From this it follows that equal matrices are of the same order but matrices of the same order are not necessarily equal.
A=B aij = bij for all i and j
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Null matrixAny matrix, all of whose elements are zero, is called a null or zero matrix and is denoted by O.
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Matrix Addition
A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B is defined by:
ijijij bac
Note: all three matrices are of the same dimension
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Addition
A a11 a12
a21 a22
B b11 b12
b21 b22
C a11 b11 a12 b12
a21 b21 a 22 b22
If
and
then
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Matrix Addition Example
A B 3 4
5 6
1 2
3 4
4 6
8 10
C
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Multiplication by a scalar
If A is a given matrix and a scalar then
A is the matrix each of whose elements is
times the corresponding element of A.
Thus A
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The Identity
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Identity Matrix
I
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Square matrix with ones on the diagonal and zeros elsewhere.
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Equal Matrices
Two matrices A and B are said to be equal if, and only if, each element aij of A is equal to the corresponding element bij of B.
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The Null matrix
Any matrix all of whose elements are zero is called a null or zero matrix
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Transpose Matrix
A'
a11 a21 ,, am1
a12 a22 ,, am 2
a1n a2n ,, amn
Rows become columns and columns become rows
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Square Matrix
B
5 4 7
3 6 1
2 1 3
Same number of rows and columns
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Matrix Subtraction
C = A - BIs defined by
Cij Aij Bij
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Let A and B be two matrices. If the number of columns in A is equal to the number of rows in B we say that A and B are conformable for the matrix product AB.
If A is order m×n and B is of order n×p, then the product AB is defined and is a matrix of order m×p.
Matrix Multiplication
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Matrix Multiplication
Matrices A and B have these dimensions:
[r x c] and [s x d]
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Matrix Multiplication
Matrices A and B can be multiplied if:
[m x n] and [n x p]
n = n
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Matrix Multiplication
The resulting matrix will have the dimensions:
[m x n] and [n x p]
m x p
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Computation: A x B = C
A a11 a12
a21 a22
B b11 b12 b13
b21 b22 b23
232213212222122121221121
2312131122121211 21121111
babababababa
babababababaC
[2 x 2]
[2 x 3]
[2 x 3]
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Computation: A x B = C
A
2 3
1 1
1 0
and B
1 1 1
1 0 2
[3 x 2] [2 x 3]A and B can be multiplied
1 1 1
3 1 2
8 2 5
12*01*1 10*01*1 11*01*1
32*11*1 10*11*1 21*11*1
82*31*2 20*31*2 51*31*2
C
[3 x 3]
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Computation: A x B = C
1 1 1
3 1 2
8 2 5
12*01*1 10*01*1 11*01*1
32*11*1 10*11*1 21*11*1
82*31*2 20*31*2 51*31*2
C
A
2 3
1 1
1 0
and B
1 1 1
1 0 2
[3 x 2] [2 x 3]
[3 x 3]
Result is 3 x 3
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Note:
If A is an m×n and B is n×p matrix, then AB is an m×p matrix. Hence we see that BA is defined only when p=m.
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Inversion
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The Inverse of a Matrix
Let A be a square matrix. A matrix B such that AB=I=BA is called the inverse matrix of A and is denoted by A-1.
So if A-1 exists, we have AA-1=I=A-1A and the matrix is said to be invertible.
If a matrix has no inverse, then it is said to be non-invertible.
Definition:
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The Inverse of a Matrix
IAAAA 11
Like a reciprocal in scalar math
Like the number one in scalar math
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Linear System of Simultaneous Equations
1st Precinct : x1 x2 6
2nd Pr ecinct : 2x1 x2 9
First precinct: 6 arrests last week equally divided between felonies and misdemeanors.
Second precinct: 9 arrests - there were twice as many felonies as the first precinct.
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Solution
9
6 *
1 2
1 1
2
1
x
x
3
3
2
1
x
x
1 2
1 1 Note: Inverse of is
1 2
1 1
9
6*
1 2
1 1 *
1 2
1 1*
1 2
1 1
2
1
x
x Premultiply both sides by inverse matrix
3
3 *
1 0
0 1
2
1
x
x A square matrix multiplied by its inverse results in the identity matrix.
A 2x2 identity matrix multiplied by the 2x1 matrix results in the original 2x1 matrix.
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aijxj bi or Ax bj1
n
x A 1Ax A 1b
n equations in n variables:
unknown values of x can be found using the inverse of matrix A such that
General Form
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Garin-Lowry Model
Ax y x
y Ix Ax
y (I A)x
(I A) 1 y x
The object is to find x given A and y . This is done by solving for x :
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Matrix Operations in Excel
Select the cells in which the answer will appear
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Matrix Multiplication in Excel
1) Enter “=mmult(“
2) Select the cells of the first matrix
3) Enter comma “,”
4) Select the cells of the second matrix
5) Enter “)”
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Matrix Multiplication in Excel
Enter these three key strokes at the same time:
control
shift
enter
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Matrix Inversion in Excel
Follow the same procedure Select cells in which answer is to be
displayed Enter the formula: =minverse( Select the cells containing the matrix to be
inverted Close parenthesis – type “)” Press three keys: Control, shift, enter
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