# Matrices and Matrix Solutions - University of Florida Matrices and Matrix Solutions Solving Equations With Matrices

Fundamental Concepts Types of Matrices

Matrix Types, Continued Matrix Types, Continued

Matrix Transpose Process of interchanging rows and columns Matrix Equality

I. COMMUTATIVE PROPERTY: A+B=B+A II. ASSOCIATIVE PROPERTY:

A + (B+C) = (A + B) + C NOTE: Thus order of operation in addition is not important!

Matrix Multiplication Matrix Multiplication Properties

Representing Equations Using Matrices Solving Equations By Matrix

Inverse Example of Matrix Solution

Inverse of a 2x2 Matrix a b A

c d

1 d b A da bc c a 1

Note: the determinant of A is: A da bc

Steps: 1. Flip the diagonals, 2. Negate the off-diagonals, and

3. Divide by the determinant. 4. Matrix is SINGULAR when determinant = 0, and thus has no inverse. Other Methods of Matrix Solution

Inverse by elementary row transformations Gaussian elimination Others

For large matrices with many zeros, sparse matrix methods can be used Spreadsheet Functions

TRANSPOSE MMULT MINVERSE The trick in Excel is to press

simultaneously