Matrix Algebra Basics Chapter 3 Section 5

Algebra

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

Row Matrix [1 x n] matrix

Column Matrix [m x 1] matrix

Square Matrix Same number of rows and columns

The Identity

Identity Matrix Square matrix with ones on the diagonal and zeros elsewhere.

Transpose Matrix Rows become columns and columns become rows

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 IfIf and then

Matrix Addition Example

Matrix addition: Ex 2: (Matrix addition)

Matrix Subtraction C = A - B Is defined by

Matrix subtraction: Ex 3: (Scalar multiplication and matrix subtraction) Find (a) 3A, (b) –B, (c) 3A – B

(a)(a) (b) (b) (c)(c) Solution:

Then (1) A+B = B + A (2) A + ( B + C ) = ( A + B ) + C (3) ( cd ) A = c ( dA ) (4) 1A = A (5) c( A+B ) = cA + cB (6) ( c+d ) A = cA + dA Properties of matrix addition and scalar multiplication:

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] r x d

Computation: A x B = C [2 x 2] [2 x 3]

Computation: A x B = C [3 x 2][2 x 3] A and B can be multiplied [3 x 3]

Computation: A x B = C [3 x 2][2 x 3] [3 x 3] Result is 3 x 3

Inversion

Matrix Inversion Like a reciprocal in scalar math Like the number one 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.

Solution Note: Inverse ofis 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.

n equations in n variables: unknown values of x can be found using the inverse of matrix A such that General Form

Garin-Lowry Model The object is to find x given A and y. This is done by solving for 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