determin.ant — Linear Algebra from Scratch

Where you'll see this: inverting a camera matrix to go from screen coordinates back to world coordinates. Decryption in cryptography (inverting the encryption matrix). Solving

Ax = b

efficiently by computing

x = A^{-1}b

. Any time you need to "undo" a transformation — you need the inverse.

A transformation is invertible if there's another transformation that undoes it. Apply A, then apply $A^{-1}$ , and you're back where you started.

The inverse undoes the transformation

For a 2×2 matrix, the inverse has a closed-form formula:

A^{-1} = \frac{1}{\det(A)}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}

Notice the $\det(A)$ in the denominator. If det = 0, the formula breaks — division by zero. There is no inverse.

Seeing it in the interactive

Use the buttons above the grid to cycle through three views:

Show A — the original transformation
Show A⁻¹ — the inverse: notice it "unscrambles" A's grid
Show A · A⁻¹ — their product, which is always the identity

Try the Singular preset. The inverse buttons grey out — no inverse exists. Then try Near-singular: technically invertible, but notice how the inverse entries explode in magnitude. That's numerical instability in action.

The product A · A⁻¹ = I

The identity matrix $I$ is the "do nothing" transformation. $A \cdot A^{-1} = I$ means applying A then $A^{-1}$ leaves every vector unchanged — the grid returns to its original position.

Invertibility summarised

$\det(A) \neq 0$ ↔ A is invertible ↔ system $Ax = b$ has a unique solution
$\det(A) = 0$ ↔ A is singular ↔ A collapses space ↔ no inverse

These are all the same fact, viewed from different angles. The determinant is the single number that tells you which world you're in.