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

Matrix Addition

Matrix Addition Properties 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 Note: the determinant of A is: 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 <Ctrl><Shift><Enter> simultaneously