Matrix Algebra Basics

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Matrix Algebra Basics

• Pam Perlich
• Urban Planning 5/6020

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Algebra
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Matrix
A matrix is any doubly subscripted array of
elements arranged in rows and columns.
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Row Vector
1 x n matrix
 
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Column Vector
m x 1 matrix
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Square Matrix
Same number of rows and columns
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The Identity
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Identity Matrix
Square matrix with ones on the diagonal and zeros
elsewhere.
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Transpose Matrix
Rows become columns and columns become rows
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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  
Note all three matrices are of the same dimension
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Addition
If
and
then
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Matrix Addition Example
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Matrix Subtraction
C A - B Is defined by
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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
r x c and s x d
c s
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Matrix Multiplication
The resulting matrix will have the dimensions
r x c and s x d
r x d
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Computation A x B C
2 x 2
2 x 3
2 x 3
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Computation A x B C

3 x 2
2 x 3
A and B can be multiplied
3 x 3
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Computation A x B C

3 x 2
2 x 3
Result is 3 x 3
3 x 3
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Inversion
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Matrix Inversion
Like a reciprocal in scalar math
Like the number one in scalar math
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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.
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Solution
Premultiply both sides by inverse matrix
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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General Form
n equations in n variables
unknown values of x can be found using the
inverse of matrix A such that

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Garin-Lowry Model
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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