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

1. Matrix Algebra Basics
Pam Perlich
Urban Planning 5/6020

2. Al gebra

3. Matrix
A =
a11 ,…, a1n
a21 ,…, a2n
… … … …
am1 ,…, amn










= Aij{ }
A matrix is any doubly subscripted array of
elements arranged in rows and columns.

4. Row Vector
[1 x n] matrix
[ ] { }jn aaaaA ,,21 =…

5. Column Vector
{ }i
m
a
a
a
a
A
2
1
=












…
=
[m x 1] matrix

6. Square Matrix
B =
5 4 7
3 6 1
2 1 3








Same number of rows and columns

7. The
Ident i t y

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

12. Matrix Addition Example
A + B =
3 4
5 6




+
1 2
3 4



 =
4 6
8 10



 = C

13. Matrix Subtraction
C = A - B
Is defined by
Cij{ } = Aij{ } − Bij{ }

14. Matrix Multiplication
Matrices A and B have these dimensions:
[r x c] and [s x d]

15. Matrix Multiplication
Matrices A and B can be multiplied if:
[r x c] and [s x d]
c = s

16. Matrix Multiplication
The resulting matrix will have the dimensions:
[r x c] and [s x d]
r x d

17. Computation: A x B = C
A =
a11 a12
a21 a22




B =
b11 b12 b13
b21 b22 b23










+++
+++
=
232213212222122121221121
231213112212121121121111
babababababa
babababababa
C
[2 x 2]
[2 x 3]
[2 x 3]

18. 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










=










=+=+=+
=+=+=+
=+=+=+
=
111
312
825
12*01*110*01*111*01*1
32*11*110*11*121*11*1
82*31*220*31*251*31*2
C
[3 x 3]

19. Computation: A x B = C










=










=+=+=+
=+=+=+
=+=+=+
=
111
312
825
12*01*110*01*111*01*1
32*11*110*11*121*11*1
82*31*220*31*251*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

20. Inversi on

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.

23. Solution






=











9
6
*
12
11
2
1
x
x






=





3
3
2
1
x
x






12
11
Note: Inverse of is 





−
−
12
11
9
6
*
12
11
*
12
11
*
12
11
2
1












−
−
=

















−
−
x
x Premultiply both sides by
inverse matrix
3
3
*
10
01
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.

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 :

26. Matrix Operations in Excel
Select the
cells in
which the
answer
will
appear

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