Matrix Multiplication
Row-by-column products; not commutative.
Key idea
Row-by-column products; not commutative.
In depth
The (i, j) entry of AB is the dot product of row i of A with column j of B. It requires the columns of A to equal the rows of B. It represents composition of linear maps, so the order matters.
Worked example
[1 2; 3 4][5 6; 7 8] = [19 22; 43 50].
Formula
Real-world uses
- Applying a series of transformations in 3D graphics
- Computing outputs of neural network layers
Common mistake
Multiplying entry by entry, or assuming AB = BA.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Matrix Multiplication?
Row-by-column products; not commutative. It is a topic at the advanced level in Linear Algebra.
What is the formula for Matrix Multiplication?
The key formula is (AB)ᵢⱼ = Σ aᵢₖ bₖⱼ.
Where is Matrix Multiplication used in real life?
Real-world examples: Applying a series of transformations in 3D graphics; Computing outputs of neural network layers.
What should I learn before Matrix Multiplication?
Make sure you are comfortable with Matrix Addition and Scaling, the previous topic in Linear Algebra.
What is a common mistake with Matrix Multiplication?
Multiplying entry by entry, or assuming AB = BA.
How can I practise Matrix Multiplication?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Matrix Multiplication?
Move on to Identity Matrix next.
Related advanced topics in Linear Algebra
- Dot Product – Sum of component products.
- Angle Between Vectors – Cosine from the dot product.
- Orthogonality – Vectors are orthogonal when their dot product is 0.
- Projection – Component of one vector along another.
- Cross Product – A vector perpendicular to two vectors in 3D.
- Lines and Planes in Vector Form – Parametric and normal equations.