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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. Row Vector. [1 x n] matrix. Column Vector. [m x 1] matrix. Square Matrix. Same number of rows and columns. The Identity.

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matrix algebra basics

Matrix Algebra Basics

Pam Perlich

Urban Planning 5/6020

matrix
Matrix

A matrix is any doubly subscripted array of elements arranged in rows and columns.

row vector
Row Vector

[1 x n] matrix

column vector
Column Vector

[m x 1] matrix

square matrix
Square Matrix

Same number of rows and columns

identity matrix
Identity Matrix

Square matrix with ones on the diagonal and zeros elsewhere.

transpose matrix
Transpose Matrix

Rows become columns and columns become rows

matrix addition and subtraction
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

addition
Addition

If

and

then

matrix subtraction
Matrix Subtraction

C = A - B

Is defined by

matrix multiplication
Matrix Multiplication

Matrices A and B have these dimensions:

[r x c] and [s x d]

matrix multiplication15
Matrix Multiplication

Matrices A and B can be multiplied if:

[r x c] and [s x d]

c = s

matrix multiplication16
Matrix Multiplication

The resulting matrix will have the dimensions:

[r x c] and [s x d]

r x d

computation a x b c
Computation: A x B = C

[2 x 2]

[2 x 3]

[2 x 3]

computation a x b c18
Computation: A x B = C

[3 x 2]

[2 x 3]

A and B can be multiplied

[3 x 3]

computation a x b c19
Computation: A x B = C

[3 x 2]

[2 x 3]

Result is 3 x 3

[3 x 3]

matrix inversion
Matrix Inversion

Like a reciprocal in scalar math

Like the number one in scalar math

linear system of simultaneous equations
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.

solution

Note: Inverse of

is

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.

general form
General Form

n equations in n variables:

unknown values of x can be found using the inverse of matrix A such that

garin lowry model
Garin-Lowry Model

The object is to find x given A and y . This is done by solving for x :

matrix operations in excel
Matrix Operations in Excel

Select the cells in which the answer will appear

matrix multiplication in excel
Matrix Multiplication in Excel
  • Enter “=mmult(“
  • Select the cells of the first matrix
  • Enter comma “,”
  • Select the cells of the second matrix
  • Enter “)”
matrix multiplication in excel28
Matrix Multiplication in Excel

Enter these three key strokes at the same time:

control

shift

enter

matrix inversion in excel
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