Matrix Inverses and Determinants

The inverse lets you “undo” a matrix transformation, and the determinant tells you whether that’s even possible. In this lesson you’ll learn how to compute both for $2 \times 2$ matrices and understand what they mean geometrically.

The inverse of a matrix $A$ (written $A^{-1}$) satisfies:

$$A \cdot A^{-1} = A^{-1} \cdot A = I$$

where $I$ is the identity matrix (1s on the diagonal, 0s elsewhere). Only square matrices can have inverses, and not all of them do.

The determinant of a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is:

$$\det(A) = ad - bc$$

What it tells you:

• $\det(A) \neq 0$: $A$ is invertible
• $\det(A) = 0$: $A$ is singular (no inverse exists)
• The absolute value $\lvert \det(A) \rvert$ is the area scaling factor of the transformation
• A negative determinant means the transformation flips orientation

If $\det(A) \neq 0$:

$$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

Swap $a$ and $d$, negate $b$ and $c$, divide by the determinant.

The matrix used in the diagram is:

$$A = \begin{bmatrix} 2 & 1 \\ 0 & 1.5 \end{bmatrix}$$

Its determinant: $\det(A) = (2)(1.5) - (1)(0) = 3$

The blue unit square has area 1. After applying $A$, the green parallelogram has area $= \lvert \det(A) \rvert = 3$. The determinant is literally the area scaling factor of the transformation: whatever shape goes in, its area gets multiplied by $\lvert \det(A) \rvert$.

Here, $A^{-1}$ maps the green parallelogram back to the original blue unit square. The inverse “undoes” the transformation completely.

Using the 2x2 inverse formula on $A = \begin{bmatrix} 2 & 1 \\ 0 & 1.5 \end{bmatrix}$:

$$A^{-1} = \frac{1}{3} \begin{bmatrix} 1.5 & -1 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} 0.5 & -\frac{1}{3} \\ 0 & \frac{2}{3} \end{bmatrix}$$

Multiplying any point on the green parallelogram by $A^{-1}$ sends it back to the corresponding point on the original unit square. That’s the power of the inverse: it reverses the transformation exactly.

Example 1: Computing the Inverse

$A = \begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix}$

$\det(A) = (4)(1) - (3)(2) = 4 - 6 = -2$

Since $\det(A) \neq 0$, $A$ is invertible:

$$A^{-1} = \frac{1}{-2} \begin{bmatrix} 1 & -3 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} -0.5 & 1.5 \\ 1 & -2 \end{bmatrix}$$

Verify: $A \cdot A^{-1} = \begin{bmatrix} 4(-0.5)+3(1) & 4(1.5)+3(-2) \\ 2(-0.5)+1(1) & 2(1.5)+1(-2) \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ ✓

Example 2: Determinant as Area Scaling

If $\det(A) = 5$, applying $A$ stretches areas by a factor of 5.

If $\det(A) = -3$, it stretches areas by 3 and flips orientation (mirrors the space).

Example 3: Singular Matrix

$A = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix}$

$\det(A) = (2)(2) - (4)(1) = 0$

No inverse exists. The rows are linearly dependent (row 1 = 2 $\times$ row 2), so $A$ squashes 2D space down to a line.

• Graphics and games: the inverse of a transformation matrix converts from world space back to object space (essential for collision detection and mouse picking)
• Physics: inverses solve for unknown forces when working with mass matrices
• Machine learning: least squares and normal equations rely on matrix inverses
• Cryptography: some encryption schemes use matrix inverses over finite fields

Example: When you click on a 3D object in a game, the engine uses the inverse of the model-view-projection matrix to trace a ray from your mouse position back into the 3D scene and figure out which object you hit.