The Inverse of a Matrix

The inverse of a matrix is simply the transformation that undoes it, the same way dividing by 3 undoes multiplying by 3. It's good for running a process backward: decoding a message, or working out what input produced the output you see. The $2\times2$ formula looks odd, but it's just three small moves: swap the two diagonal entries $a$ and $d$, flip the signs of $b$ and $c$, and divide everything by $ad - bc$ (the determinant, a number that says how much $A$ stretches area). If that number is zero, there's nothing to divide by, and there's no inverse. That's the whole formula.

Solving systems. You know that $3x = 6$ is solved by dividing by 3. Matrices work the same way: if $A$ is invertible, $A\vec x = \vec b$ has exactly one solution, $\vec x = A^{-1}\vec b$. (Computers usually row reduce instead, because it's faster and more accurate.)

Finding inverses. Row reduce $[A \mid I]$. If the left side becomes $I$, the right side is $A^{-1}$. If a row of zeros appears on the left, $A$ isn't invertible. Think about why this works: the row operations that turn $A$ into $I$ are exactly what $A^{-1}$ does, and applying them to $I$ writes that down for you.

Rules. $(AB)^{-1} = B^{-1}A^{-1}$, and $(A^T)^{-1} = (A^{-1})^T$.

A common mistake. It's tempting to write $(AB)^{-1} = A^{-1}B^{-1}$, because that's how it works for numbers. It isn't: $AB$ means "do $B$, then $A$," so undoing it means undoing $A$ first, then $B$. It's like putting on socks and then shoes. To undo that, the shoes come off first.

Undoing a camera's projection to recover 3D positions, decoding error-correcting codes, and inverting a known blur all rely on matrix inverses (or solving the equivalent systems).