Matrix Addition and Scalar Multiplication
Combining matrices element-by-element and scaling every entry uniformly.
Introduction
Combining matrices element-by-element and scaling every entry uniformly.
Past Knowledge
You understand matrix notation and dimensions ().
Today's Goal
We add matrices entry-by-entry and multiply every entry by a scalar constant.
Future Success
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)
(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:
is
is
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)
Practice Quiz
Practice Quiz
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