MISSION · Algebra
Matrices as Transformations
A grid of numbers that moves every point at once
LEVEL 1 · EXPLORE
Make the idea move
Change the controls and watch what happens. Follow your curiosity and see what you notice.
Loading the activity…
LEVEL 2 · LEARN
Unlock the explanation
Open the chapters in any order. Try the examples as you go.
The idea in one sentence
A matrix is a table of numbers, and a \(2 \times 2\) one is best understood as an instruction that stretches, rotates, or flips the entire plane in a single step.
Where the columns go
Take any \(2 \times 2\) matrix:
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}\]The first column is where the point \((1, 0)\) lands. The second is where \((0, 1)\) lands. That’s it — and because every other point is a combination of those two, knowing their destinations determines everything.
\(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) sends \((1,0)\) to \((0,1)\) and \((0,1)\) to \((-1,0)\): a quarter turn anticlockwise. Read that way, matrices stop being arbitrary tables and become descriptions you can picture.
The identity matrix \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) leaves both where they were, so it changes nothing — the matrix equivalent of multiplying by 1.
The determinant is an area
\[\det = ad - bc\]The unit square, after the transformation, becomes a parallelogram, and the determinant is its area. A determinant of 3 means every region triples; a determinant of 1 means areas are preserved, as under any rotation.
A negative determinant means the plane has been flipped over — orientation reversed, like a reflection.
A determinant of zero is the interesting case. The square has been squashed flat onto a line, all area destroyed. Information is lost: many different points now land on the same place, so the transformation cannot be undone. That’s exactly when the matrix has no inverse, and it’s why the determinant sits in the denominator of the inverse formula.
Multiplication is composition
Matrix multiplication looks like an arbitrary ritual of rows against columns until you know what it’s for: the product \(AB\) is the single matrix that does \(B\) first, then \(A\).
That explains its two famous oddities:
- Order matters. \(AB \ne BA\) in general — rotating then reflecting is not the same as reflecting then rotating. Try it with your hands.
- The dimensions must agree. The output of the first has to be a valid input for the second.
It also explains why matrices are everywhere in computing. Every rotation, scale, and camera angle in a 3D scene is a matrix, and combining a chain of them into one multiplication is what makes the whole thing fast enough to render.
Solving systems
A pair of simultaneous equations is one matrix equation:
\[\begin{pmatrix} 2 & 3 \\ 1 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 12 \\ 1 \end{pmatrix}\]which is \(A\mathbf{x} = \mathbf{b}\), solved by \(\mathbf{x} = A^{-1}\mathbf{b}\).
And the geometry now predicts the algebra. A zero determinant means no inverse, which means the system has no unique solution — precisely the parallel-or-identical-lines case. Two unrelated-looking facts turn out to be the same fact.
Common mix-ups
- Assuming \(AB = BA\). It usually isn’t.
- Multiplying element by element. Rows against columns, summing products.
- Forgetting the determinant can be negative or zero, both of which mean something.
- Trying to invert a singular matrix. No inverse exists when \(\det = 0\).
Check yourself
What does \(\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}\) do to the plane, and what is its determinant?
It stretches everything twice as wide and three times as tall. The determinant is 6, so every area is six times bigger.
LEVEL 3 · REFLECT
Add this quest to your trail
When you feel ready, mark this idea as explored. You can always come back and try it again.
EXPLORER STRATEGIES