Inverse & Invertibility

The last lesson told you whether a matrix could be undone — a non-zero determinant was the green light. This one builds the undoing. Picture A as something that takes every point of space to a new spot, rotating, stretching, shearing. The inverse A⁻¹ is the transformation that puts every point back exactly where it began, like rewinding a video.

It exists only when A threw nothing away on the trip out.

Undoing a transformation

The inverse of A is the matrix A⁻¹ for which applying one and then the other leaves you where you started:

A·A⁻¹ = A⁻¹·A = I        (I = the identity, the do-nothing matrix)

This works only if A destroyed no information. If A flattened the plane onto a line — collapsing area to zero — the lost dimension can never be recovered, so no inverse exists. That is precisely the det(A) = 0 case from last lesson: a matrix is invertible exactly when it keeps space “full”. Stretch and rotate the matrix below, then picture the transformation that would push the parallelogram back to a square — that is what A⁻¹ does. Collapse it to a line and nothing can undo it.

The 2×2 formula

For a 2×2 matrix there is a formula worth memorising:

• Swap the diagonal entries.
• Negate the off-diagonal ones.
• Divide by the determinant.

The determinant ad − bc sits in the denominator — which is exactly why a zero determinant has no inverse: you would be dividing by zero. The thread from last lesson ties off here.

It is worth saying what the formula is not, because the guess nearly everyone makes first is the wrong one: the inverse is not the reciprocal of each entry. Take A = [[1, 2], [3, 4]]. Flipping every entry gives [[1, 0.5], [1/3, 0.25]], while the true inverse is [[−2, 1], [1.5, −0.5]] — different in every position, and carrying minus signs that no reciprocal could ever produce.

Undoing a transformation is a statement about the matrix acting as a whole, and its entries do not get undone one at a time. (The lone exception proves the rule: a diagonal matrix does invert entrywise, diag(2, 5)⁻¹ = diag(1/2, 1/5), precisely because its entries never mix with each other.)

One property, many faces

For an n×n matrix A, the following statements are either all true or all false together — different views of the single property of invertibility:

In words, the equivalence chain is:

• A is invertible.
• det(A) ≠ 0.
• A has full rank (its n columns between them still span the whole n-dimensional space — nothing has collapsed).
• Its columns are independent.
• 0 is not an eigenvalue.
• Ax = 0 has only the trivial solution x = 0.

Spot any one of these in a question and you have all the others for free. One more rule, the mirror of det(AB) = det(A)det(B): the inverse of a product reverses order, (AB)⁻¹ = B⁻¹A⁻¹ — like taking off socks after shoes.

A worked example

Invert A = [[2, 1], [1, 1]] and verify it.

det(A) = (2)(1) − (1)(1) = 2 − 1 = 1        non-zero → the inverse exists

swap the diagonal, negate the off-diagonal, divide by det = 1:
A⁻¹ = (1/1) · [[ 1, −1],   =  [[ 1, −1],
               [−1,  2]]        [−1,  2]]

verify A·A⁻¹:
[[2,1],[1,1]] · [[1,−1],[−1,2]] = [[2−1, −2+2], [1−1, −1+2]] = [[1,0],[0,1]] = I  ✓

The product is the identity, so the inverse is correct.

A question to carry forward

A general matrix shoves vectors around — rotating, shearing, turning them off their original line. But for almost every matrix there are a few special directions it leaves on their own line, merely stretching or shrinking them. Here is the thread onward: which directions does a matrix only stretch, by how much, and why does that pair — direction and stretch — unlock so much of what a matrix is?

In one breath

• The inverse A⁻¹ undoes A: A·A⁻¹ = A⁻¹·A = I; it exists only if A kept space full (threw no dimension away).
• 2×2 formula: A⁻¹ = 1/(ad−bc) · [[d, −b], [−c, a]] — swap the diagonal, negate the off-diagonal, divide by the determinant.
• The determinant is in the denominator, so det(A) = 0 (singular) → no inverse.
• The chain: invertible ⇔ det≠0 ⇔ full rank ⇔ columns independent ⇔ 0 not an eigenvalue ⇔ Ax=0 only trivially. Recognise one, get all.
• Product reverses: (AB)⁻¹ = B⁻¹A⁻¹ (mirror of the determinant product rule).

Practice