Matrix Inverse

The 2× 2 inverse formula, invertibility via A ≠ 0, and solving Ax = b.

The idea

The inverse of a square matrix $A$ is the matrix that undoes $A$: applying $A$ and then its inverse returns every vector to where it started.

Definition (Inverse matrix).

The inverse of a square matrix $A$, written $A^{-1}$, is the matrix with $A A^{-1} = A^{-1} A = I,$ where $I$ is the identity matrix — ones down the diagonal, zeros elsewhere — the matrix that leaves every vector unchanged.

The inverse solves a whole system at once: multiplying $A\vec{x} = \vec{b}$ on the left by $A^{-1}$ gives $\vec{x} = A^{-1}\vec{b}$.

Not every square matrix has an inverse. A matrix that flattens the plane onto a line has discarded information, and no matrix can recover it. The determinant detects exactly this failure.

Theorem (Invertibility and the 2×2 inverse).

A square matrix $A$ has an inverse if and only if $\det A \neq 0$. For a $2 \times 2$ matrix, $\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.$

The determinant $ad - bc$ in the denominator shows the formula failing exactly when the inverse fails to exist.

Ways to work on it

• Walkthrough. Apply the 2× 2 inverse formula and the invertibility test.
• Practice. Compute then inverse entries for a random invertible 2× 2.
• Hardest. Solve a linear system with the inverse, and invert a product of matrices.

Not sure where to start? Take the ten-question placement test.