Systems of Linear Equations; Matrices

Introduction to Matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. They are fundamental tools in algebra for representing and solving systems of linear equations, as well as for modeling real-world applications in business, economics, and sciences.

• Matrix Size: Defined by the number of rows and columns (e.g., a 2x3 matrix has 2 rows and 3 columns).

• Notation: Matrices are usually denoted by uppercase letters (A, B, C, etc.).

• Application: Used to organize data, perform calculations, and solve systems efficiently.

Matrices: Basic Operations

Matrix Addition and Subtraction

Matrix addition and subtraction are defined only for matrices of the same size. The operations are performed element-wise, meaning each corresponding element is added or subtracted.

• Equality: Two matrices are equal if they have the same size and all corresponding elements are equal.

• Addition: for all .

• Subtraction: for all .

• Undefined Operations: Addition or subtraction is not defined for matrices of different sizes.

Example: If and , then:

Matrix Properties

Matrix addition has several important properties, provided the matrices are of the same size:

• Commutative Property:

• Associative Property:

Zero Matrix

A zero matrix is a matrix in which all elements are zero. It acts as the additive identity in matrix addition.

• Notation: Often denoted simply as "0" for any size.

• Property: for any matrix A.

Negative of a Matrix

The negative of a matrix , denoted , is formed by taking the negative of each element in $M$.

• Definition: If , then .

• Property:

Subtraction of Matrices

Matrix subtraction is defined as adding the negative of one matrix to another:

• Formula:

Multiplication of a Matrix by a Number (Scalar Multiplication)

Multiplying a matrix by a scalar (number) means multiplying each element of the matrix by that number.

• Formula:

• Example: If and , then

Applications of Matrix Operations

Sales Commissions Example

Matrices can be used to model real-world scenarios such as sales data and commissions. For example, matrices can represent sales figures for different products and months, and matrix operations can be used to calculate combined sales, increases, or commissions.

• Addition: Combine sales data from two months.

• Subtraction: Find the increase in sales from one month to another.

• Scalar Multiplication: Calculate commissions by multiplying sales by the commission rate.

Matrix Multiplication

Product of a Row Matrix and a Column Matrix

The product of a row matrix and a column matrix is defined if the number of elements in the row equals the number in the column. The result is a single number (scalar), calculated as the sum of products of corresponding elements.

• Formula:

General Matrix Product

If is an matrix and is an matrix, the product is an matrix. The element in the th row and th column is the sum of products of the $i$th row of $A$ and the $j$th column of $B$.

• Formula: , where

• Existence: The product is defined only if the number of columns in equals the number of rows in .

Size of Matrix Product

Before multiplying matrices, check their sizes:

• If is and is , then is .

• If the sizes do not match, the product is not defined.

Matrix Multiplication Examples

Several examples demonstrate how to multiply matrices, check for existence of products, and interpret the results. These include both numeric and application-based examples, such as labor cost calculations.

• Example: Labor costs for manufacturing skis can be calculated by multiplying matrices representing labor hours and wage rates.

Solving Matrix Equations

Matrix Equations and Systems

Matrix equations can be used to solve for unknowns in systems of equations. By equating matrices, you can set up and solve multiple equations simultaneously.

• Method: Equate corresponding elements to form a system of equations.

• Solution: Use substitution, elimination, or matrix methods (such as Gauss-Jordan elimination) to solve.

Summary Table: Matrix Operations

Additional info: These notes expand on the original slides by providing full definitions, formulas, and examples for each operation, as well as a summary table for quick reference. Images included are directly relevant to matrix addition and calculator technology as shown in the slides.