The Inverse Matrix

Every editing app has an Undo button. Drag a shape, rotate it, stretch it — and if you don't like the result, one click puts it back exactly where it started. Matrices need an undo button too. If a matrix A rotates and stretches a picture, its inverse A^{-1} is the matrix that un-rotates and un-stretches it, landing every point back on its original spot.

Apply A and then A^{-1} (in either order) and you return to exactly where you started — the identity, the matrix version of "do nothing":

A^{-1}A = AA^{-1} = I.

That's the whole definition. A^{-1} is defined to be whichever matrix, multiplied by A on either side, gives back I — nothing more. It plays exactly the role \tfrac13 plays for the number 3: multiplying by a number and then by its reciprocal gets you back to 1; applying a transformation and then its inverse gets you back to "no transformation at all."

The one condition that decides everything

Not every matrix has an undo button. A matrix A is invertible — has an A^{-1} at all — exactly when its determinant is non-zero:

A^{-1} \text{ exists} \iff \det A \neq 0.

Remember what the determinant measures: how much a transformation scales area (in 2D) or volume (in 3D). If \det A = 0, the transformation squashes the whole plane flat onto a line — or onto a single point. Area 2 shrinks to area 0. Once two different starting points have been squeezed onto the very same spot, no matrix on Earth can look at that spot and tell you which point it used to be — the information is simply gone. An inverse would have to un-squash a line back into a full plane, stretching one point into two, and no function can do that. So "singular" (det = 0) and "not invertible" mean exactly the same thing.

Warp, then un-warp

Watch it happen to an entire grid, not just one point. Step through the stages below: first the grid sits at rest. Apply A and it stretches and skews into a slanted parallelogram grid. Then apply A^{-1} to that warped grid, and it snaps perfectly back to the original square grid — every line, every angle, restored. That round trip, warp then un-warp, is what A^{-1}A = I looks like in motion.

Worked example: checking a claimed inverse

Suppose someone hands you a matrix and claims it's the inverse of another. You don't have to trust them — just multiply and see if you land on I. Take

A = \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}, \qquad \text{claimed: } A^{-1} = \begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}.

Multiply A \cdot A^{-1}, one row-times-column at a time:

A A^{-1} = \begin{bmatrix} 2(1) + 1(-1) & 2(-1) + 1(2) \\ 1(1) + 1(-1) & 1(-1) + 1(2) \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I.

It checks out — but the definition demands both orders agree, so verify A^{-1}A too:

A^{-1} A = \begin{bmatrix} 1(2) + (-1)(1) & 1(1) + (-1)(1) \\ -1(2) + 2(1) & -1(1) + 2(1) \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I.

Both products come out to I, so the claim is confirmed: this really is A^{-1}. (You'll meet an actual recipe for finding this inverse, rather than just checking it, on the next page.)

Worked example: solving a system by inverting

Here's why inverses matter beyond checking arithmetic. Two apples and one banana cost \$5; one apple and one banana cost \$3. What does each fruit cost? Write it as a matrix equation A\vec{x} = \vec{b}, with \vec{x} = (a, b) the unknown prices:

\begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 5 \\ 3 \end{bmatrix}.

This is exactly the A from the last example, and we already know its inverse. Multiply both sides on the left by A^{-1}. On the left, A^{-1}A collapses to I, leaving just \vec{x} behind:

\vec{x} = A^{-1}\vec{b} = \begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}\begin{bmatrix} 5 \\ 3 \end{bmatrix} = \begin{bmatrix} 1(5) + (-1)(3) \\ -1(5) + 2(3) \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}.

Apples cost \$2, bananas cost \$1. Check it: 2(2) + 1(1) = 5 and 1(2) + 1(1) = 3 — both original equations hold. Notice the whole "solve two equations in two unknowns" problem became one matrix-vector multiplication, once the inverse was in hand.

Worked example: when there is no undo button

Now try the same trick on a matrix that destroys information. Let

A = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix}, \qquad \det A = 2(2) - 4(1) = 0.

The determinant is zero, so this matrix should not be invertible — and here's the damage in action. Feed it two different input vectors, (1, 0) and (3, -1):

A\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \qquad A\begin{bmatrix} 3 \\ -1 \end{bmatrix} = \begin{bmatrix} 2(3) + 4(-1) \\ 1(3) + 2(-1) \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}.

Two genuinely different starting points, (1,0) and (3,-1), both land on the exact same output, (2,1). If somebody only showed you the output (2,1) and asked "which point did this come from?", there is no way to answer — both are equally valid, and infinitely many other points map there too. A true inverse would need to send (2,1) back to one specific place, but it can't decide between two (or infinitely many) equally good candidates. That's not a computational difficulty to push through with more effort — it's a hard logical wall, and it's exactly what \det A = 0 was warning you about.

Two mistakes account for almost every wrong answer involving inverses:

This isn't just textbook tidiness — invertibility is what lets scrambled information come back at all. An early cipher called the Hill cipher encodes a block of letters by multiplying them (as numbers) by a secret matrix. Sending the message is easy; the entire scheme only works, though, if that matrix is invertible — otherwise two different original messages could scramble into the same coded text, and even the person with the secret key couldn't tell them apart. Modern error-correcting codes (the ones that let a scratched CD or a weak phone signal still recover the right data) lean on the very same idea: build the encoding out of an invertible matrix, and decoding is guaranteed to be possible, in principle, by inverting it.

Even that humble Undo button from the very first paragraph is secretly doing linear algebra: every rotate, scale or skew you apply to a shape in a drawing program is stored as a matrix, and "Undo" just multiplies by that matrix's inverse. The whole idea of reversibility — in software, in codes, in physics — keeps coming back to this one question: does an inverse exist?

See it explained