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Learn Matrices

Rectangular grids of numbers that solve systems and transform space.

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A matrix is a rectangular array of numbers arranged in rows and columns, and it is one of the most useful objects in applied math. Matrices store systems of linear equations compactly, describe transformations like rotations and scalings, and run everything from computer graphics to machine learning. Learning them means mastering a few operations, above all matrix multiplication, which follows a rows-times-columns rule and does not commute. From there you learn determinants, inverses and row reduction, which together let you solve systems of equations. Learnr builds a matrices course at your level and teaches each operation in short interactive lessons with step-by-step feedback.

How to learn Matrices, step by step

Learn dimensions before anything else. A 2 by 3 matrix has 2 rows and 3 columns. You can multiply an m by n matrix by an n by p matrix only when the inner numbers match, and the result is m by p.
Practice multiplication by hand. Each entry of AB is a row of A dotted with a column of B. Multiply small pairs both ways and confirm that AB and BA usually differ, which is the property beginners forget most.
Write systems as matrix equations. Turn x + 2y = 5 and 3x + 4y = 6 into Ax = b. Row reduce the augmented matrix using swaps, scalings and row additions until you can read off the solution.
Compute determinants and inverses. For [[a, b], [c, d]], the determinant is ad minus bc and the inverse is 1 over that number times [[d, -b], [-c, a]]. A determinant of zero means no inverse exists.
See matrices as moves in the plane. Apply [[0, -1], [1, 0]] to the corners of a unit square and watch it rotate 90 degrees counterclockwise. Linking algebra to pictures makes later linear algebra far easier.
Keep the arithmetic sharp with daily practice. Matrix work punishes small arithmetic slips. A short Learnr set each day, with a streak to keep you coming back, builds the accuracy that long calculations demand.

What you'll learn

Add, scale and multiply matrices, checking dimensions first
Compute determinants of 2 by 2 and 3 by 3 matrices
Find inverses and use them to solve systems
Row reduce to echelon form with Gaussian elimination
Describe rotations and scalings in the plane as matrices

What's inside your Matrices course

A sample path Learnr builds for Matrices. Your real course adapts to your goal and level.

1. Matrix notation, dimensions and basic operations
2. Matrix multiplication and its properties
3. Systems of equations and row reduction
4. Determinants
5. Inverse matrices
6. Matrices as transformations

How long does it take to learn Matrices?

Basic matrix operations, determinants and inverses take about 3 to 6 weeks at 3 to 4 hours a week. Using matrices fluently for transformations and larger systems is part of a full linear algebra course, which takes a semester.

What you need before you start

Solid arithmetic with negatives and fractions, and experience solving systems of two equations in algebra. No calculus is needed.

Common mistakes to avoid

Assuming AB equals BA. Matrix multiplication is not commutative, so the order of factors must be kept exactly as written.
Multiplying entry by entry instead of rows by columns, which gives a different and usually meaningless result.
Trying to invert a matrix whose determinant is zero. Check the determinant first; a zero means the system has no unique solution.

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Matrices questions, answered

What are matrices used for in real life?

They rotate and project images in computer graphics, store weights in neural networks, model networks and Markov chains, and solve large systems in engineering and economics.

Is learning matrices the same as linear algebra?

Matrices are the main tool of linear algebra, but the full subject also covers vector spaces, eigenvalues and abstract linear maps. Learning matrix operations first is the usual on-ramp.