Matrix Algebra Basics
Pam Perlich
Urban Planning 5/6020
Algebra
Matrix
A matrix is any doubly subscripted array of
elements arranged in rows and columns.
a
11 ,, a1n
21 ,, a 2n
a
A Aij
m1 ,, amn
a
Row Vector
[1 x n] matrix
A a1 a 2 ,, an aj
Column Vector
[m x 1] matrix
a1
a 2
A ai
am
Square Matrix
Same number of rows and columns
4 7
5
B 6 1
3
1 3
2
The
Identity
Identity Matrix
Square matrix with ones on the
diagonal and zeros elsewhere.
1
0 0 0
0 1 0 0
I
0 0 1 0
0 0 0 1
Transpose Matrix
Rows become columns and
columns become rows
a
11 a 21 ,, am1
12 a 22 ,, am 2
a
A'
1n a 2n ,, amn
a
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
11 a12
A
I 21 a 22
a
f
b
11 b12
and B
21 b 22
b
11 b11 a12 b12
a
then C
21 b 21 a 22 b22
a
Matrix Addition Example
3
4 2
1 4
6
A B C
6 4
5 3 10
8
Matrix Subtraction
C = A - B
Is 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]
rxd
Computation: A x B = C
a
11 a12
A [2 x 2]
21 a 22
a
b
11 b12 b13
B [2 x 3]
21 b 22 b 23
b
a11b11 a12b21 a11b12 a12b22 a11b13 a12b23
C
a 21b11 a 22b21 a 21b12 a 22b22 a 21b13 a 22b23
[2 x 3]
Computation: A x B = C
3
2
1
1 1
A 1 and B
1
0 2
1
0
1
[3 x 2] [2 x 3]
A and B can be multiplied
2 *1 3 *1 5 2 *1 3 * 0 2 2 *1 3 * 2 8 5 2 8
C 1*1 1*1 2 1*1 1* 0 1 1*1 1* 2 3 2 1 3
1*1 0 *1 1 1*1 0 * 0 1 1*1 0 * 2 1 111
[3 x 3]
Computation: A x B = C
3
2
1
1 1
A 1 and B
1
0 2
1
0
1
[3 x 2] [2 x 3]
Result is 3 x 3
2 *1 3 *1 5 2 *1 3 * 0 2 2 *1 3 * 2 8 5 2 8
C 1*1 1*1 2 1*1 1* 0 1 1*1 1* 2 3 2 1 3
1*1 0 *1 1 1*1 0 * 0 1 1*1 0 * 2 1 111
[3 x 3]
Inversion
Matrix Inversion
1 1
B B BB I
Like a reciprocal Like the number one
in scalar math in scalar math
Linear System of Simultaneous
Equations
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.
1st Precinct : x1 x 2 6
2nd Pr ecinct : 2x1 x 2 9
1 1 11
Solution Note: Inverse of 2 1 is 2 1
1 1 x1 6
2 1 * x 9
2
11 11 x1 11 6 Premultiply both sides by
2 1 * 2 1 * x 2 1 * 9 inverse matrix
2
1 0 x1 3 A square matrix multiplied by its
0 1 * x 3 inverse results in the identity matrix.
2
x1 3 A 2x2 identity matrix multiplied by
x
the 2x1 matrix results in the original
3
2 2x1 matrix.
General Form
n equations in n variables:
n
aijxj bi or Ax b
j1
unknown values of x can be found using the inverse of
matrix A such that
1 1
x A Ax A b
Garin-Lowry Model
Ax y x
The object is to find x given A and y . This is
done by solving for x :
y Ix Ax
y (I A)x
1
(I A) y 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