Lesson 18.4

Matrix Addition and Scalar Multiplication

Combining matrices element-by-element and scaling every entry uniformly.

Introduction

Just as we add numbers and multiply them by scalars, we can do the same with matrices—but with a catch.Matrix addition requires both matrices to have the exact same dimensions.Scalar multiplication simply multiplies every entry by the same number. These operations are the building blocks for more complex matrix algebra.

Prerequisite Connection

You understand matrix notation and dimensions ().

Today's Increment

We add matrices entry-by-entry and multiply every entry by a scalar constant.

Why This Matters

These operations combine data from multiple sources—adding inventory matrices, scaling transformation matrices in graphics, or averaging data sets.

Key Concepts

Matrix Addition

Add corresponding entries. Dimensions must match exactly.

Scalar Multiplication

Multiply EVERY entry by the scalar :

Properties

(commutative)

(associative)

(distributive)

Worked Examples

Example 1: Adding Matrices (Basic)

Compute :

Add entry-by-entry:

Answer:

Example 2: Scalar Multiplication (Intermediate)

Compute :

Step 1: Compute

Step 2: Compute

Step 3: Subtract

Answer:

Example 3: Dimension Mismatch (Advanced)

Can we add these matrices?

Check dimensions:

NOT DEFINED!

Matrix addition requires identical dimensions.

Common Pitfalls

Trying to add matrices of different sizes

A matrix CANNOT be added to a matrix. Dimensions must match exactly.

Missing entries when scaling

The scalar multiplies EVERY entry—don't forget corners or the middle!

Sign errors in subtraction

means subtracting each entry of from the corresponding entry of . Watch negatives carefully.

Real-World Application

Image Processing: Brightness Adjustment

A grayscale image is just a matrix of pixel values (0-255). To brighten an image, ADD a constant matrix. To increase contrast, MULTIPLY by a scalar greater than 1. Photo editing apps use these exact operations on RGB matrices millions of times per image!

Brighten: (add 50 to every pixel)