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Matrix Algebra Basics
Pam PerlichUrban Planning 5/6020
Algebra
Matrix
A
a11 ,, a1n
a21 ,, a2n
am1 ,, amn
Aij
A matrix is any doubly subscripted array of elements arranged in rows and columns.
Row Vector
[1 x n] matrix
jn aaaaA ,, 2 1
Column Vector
i
m
a
a
aa
A 2
1
[m x 1] matrix
Square Matrix
B 5 4 73 6 12 1 3
Same number of rows and columns
The Identity
Identity Matrix
I
1 0 0 00 1 0 00 0 1 00 0 0 1
Square matrix with ones on the diagonal and zeros elsewhere.
Transpose Matrix
A'
a11 a21 ,, am1
a12 a22 ,, am 2
a1n a2n ,, amn
Rows become columns and columns become rows
Matrix Addition and Subtraction
A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B is defined by:
Cij Aij Bij
Note: all three matrices are of the same dimension
Addition
A a11 a12
a21 a22
B b11 b12
b21 b22
C a11 b11 a12 b12
a21 b21 a 22 b22
If
and
then
Matrix Addition Example
A B 3 45 6
1 23 4
4 68 10
C
Matrix Subtraction
C = A - BIs defined by
Cij Aij Bij
Matrix Multiplication
Matrices A and B have these dimensions:
[r x c] and [s x d]
Matrix Multiplication
Matrices A and B can be multiplied if:
[r x c] and [s x d]
c = s
Matrix Multiplication
The resulting matrix will have the dimensions:
[r x c] and [s x d]
r x d
Computation: A x B = C
A a11 a12
a21 a22
B b11 b12 b13
b21 b22 b23
232213212222122121221121
2312131122121211 21121111
babababababababababababa
C
[2 x 2]
[2 x 3]
[2 x 3]
Computation: A x B = C
A 2 31 11 0
and B
1 1 1 1 0 2
[3 x 2] [2 x 3]A and B can be multiplied
1 1 13 1 28 2 5
12*01*1 10*01*1 11*01*1
32*11*1 10*11*1 21*11*182*31*2 20*31*2 51*31*2
C
[3 x 3]
Computation: A x B = C
1 1 13 1 28 2 5
12*01*1 10*01*1 11*01*1
32*11*1 10*11*1 21*11*182*31*2 20*31*2 51*31*2
C
A 2 31 11 0
and B
1 1 1 1 0 2
[3 x 2] [2 x 3]
[3 x 3]
Result is 3 x 3
Inversion
Matrix Inversion
B 1B BB 1 I
Like a reciprocal in scalar math
Like the number one in scalar math
Linear System of Simultaneous Equations
1st Precinct : x1 x2 62nd 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.
Solution
96
*1 21 1
2
1
xx
33
2
1
xx
1 21 1
Note: Inverse of is
1 21 1
96
*1 21 1
*1 21 1
* 1 21 1
2
1
xx Premultiply both sides by
inverse matrix
33
* 1 00 1
2
1
xx 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.
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
Garin-Lowry Model
Ax y x
y Ix Axy (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 :
Matrix Operations in Excel
Select the cells in which the answer will appear
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 “)”
Matrix Multiplication in Excel
Enter these three key strokes at the same time:
control
shift
enter
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