Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
8. 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.
9. Transpose Matrix
A' =
a11 a21 ,…, am1
a12 a22 ,…, am2
… … … … …
a1n a2n ,…, amn
Rows become columns and columns
become rows
10. 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
11. Addition
A =
a11 a12
a21 a22
B =
b11 b12
b21 b22
C =
a11 + b11 a12 + b12
a21 + b21 a22 + b22
If
and
then
21. Matrix Inversion
B
−1
B = BB
−1
= I
Like a reciprocal
in scalar math
Like the number one
in scalar math
22. Linear System of Simultaneous
Equations
1st Precinct : x1 + x2 = 6
2nd Precinct : 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.
24. aijxj = bi or Ax = b
j=1
n
∑
x = A
−1
Ax = A
−1
b
n equations in n variables:
unknown values of x can be found using the inverse of
matrix A such that
General Form
25. 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 :
27. 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 “)”
28. Matrix Multiplication in Excel
Enter these
three
key
strokes
at the
same
time:
control
shift
enter
29. 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