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Mom's Basement University

Introduction to Matrices

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn what a matrix is, how to read and write matrices, and how to add and subtract matrices of the same size.

The Concept

Section titled “The Concept”

A matrix is a rectangular array of numbers arranged in rows and columns.

Example of a 2×3 matrix (2 rows, 3 columns):

[25−1307]\begin{bmatrix} 2 & 5 & -1 \\ 3 & 0 & 7 \end{bmatrix}
  • The size is written as rows × columns (e.g., 2×3).
  • Individual entries are called elements.

Matrix Addition and Subtraction - you can add or subtract two matrices only if they have the same dimensions. Add or subtract corresponding elements.

[31−24]+[5−302]=[8−2−26]\begin{bmatrix} 3 & 1 \\ -2 & 4 \end{bmatrix} + \begin{bmatrix} 5 & -3 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} 8 & -2 \\ -2 & 6 \end{bmatrix}

We will cover deeper matrix topics (determinants, inverses, matrix multiplication) in the future Linear Algebra section. In Algebra 2, we focus on basic operations and using small matrices to solve systems of equations.

Worked Example

Section titled “Worked Example”

1. Add two matrices

[4−16230]+[15−2−327]=[544−157]\begin{bmatrix} 4 & -1 & 6 \\ 2 & 3 & 0 \end{bmatrix} + \begin{bmatrix} 1 & 5 & -2 \\ -3 & 2 & 7 \end{bmatrix} = \begin{bmatrix} 5 & 4 & 4 \\ -1 & 5 & 7 \end{bmatrix}

2. Subtract two matrices

[7259]−[3416]=[4−243]\begin{bmatrix} 7 & 2 \\ 5 & 9 \end{bmatrix} - \begin{bmatrix} 3 & 4 \\ 1 & 6 \end{bmatrix} = \begin{bmatrix} 4 & -2 \\ 4 & 3 \end{bmatrix}

3. Can you add a 2×3 matrix and a 3×2 matrix?

No. The dimensions are different (2×3 vs 3×2), so addition is not defined.

Real-World Application

Section titled “Real-World Application”

Matrices are used in many practical areas:

  • Organizing data (spreadsheets, sales tables, inventory)
  • Computer graphics and video games (transformation matrices)
  • Solving systems of equations more efficiently
  • Economics and business (input-output models)
  • Engineering (stress and strain calculations)

Example: a store tracks sales of three products over two weeks in a matrix. Adding matrices lets them see total sales easily.

Quiz

Section titled “Quiz”
A 3×2 matrix has:
You can add two matrices only if they have:
What is [2, 5] + [3, −1]?
Individual numbers in a matrix are called:
If $A$ is a $2 \times 3$ matrix and $B$ is a $3 \times 4$ matrix, what are the dimensions of $AB$?