2.6.10 · D5Matrices & Determinants — Introduction
Question bank — Inverse of 2×2 matrix
True or false — justify
True or false: Every square matrix has an inverse.
False. Only non-singular matrices do. If the map flattens area to zero, destroying information, so no undo machine can exist.
True or false: If then sends every vector to the zero vector.
False. It collapses the plane onto a line (or a point only in the extreme case ). Most inputs land on that line as nonzero vectors — but many inputs share one output, which is why it can't be undone.
True or false: If is invertible, then so is , and .
True. The undo of the undo is the original machine. Since , satisfies the defining property of the inverse of .
True or false: .
True. From and the fact that determinants multiply, , so (which needs — consistent with being invertible).
True or false: If and are both invertible, then .
False. The order reverses: . To undo "apply then " you must undo first, then — like taking off shoes before socks.
True or false: The identity matrix is its own inverse.
True. , so . The machine that does nothing is undone by doing nothing.
True or false: If is negative, has no inverse.
False. Any nonzero determinant gives an inverse. A negative determinant just means the map also flips orientation (a mirror-flip), which is perfectly reversible.
True or false: Swapping the two rows of never changes whether it is invertible.
True. Row-swapping only flips the sign of the determinant, never turns a nonzero value into zero. Invertibility depends on , which survives a sign change.
True or false: If two columns of are identical, is singular.
True. Identical columns are parallel, so the parallelogram they span has zero area, i.e. . See Singular vs non-singular matrices.
Spot the error
A student writes . What went wrong?
They took the reciprocal of each entry. Matrix inversion is not entrywise; Matrix multiplication mixes rows and columns, so undoing it needs swap-diagonal, negate-off-diagonal, divide by det, not on each slot.
A student computes the adjugate of as . Which rule did they flip?
They negated the diagonal and left the off-diagonal alone — exactly backwards. The rule is swap on the main diagonal and negate off it: .
A student says ", so ." Fix it.
They multiplied by the det instead of dividing. The formula is , so it should be .
A student solves by writing . What's the mistake?
The multiplication order (and hence dimensions) is wrong. You must left-multiply: gives , with the matrix on the left of the column vector. See Solving linear systems.
A student claims ", so must equal or ." Counter it.
Not necessarily. Any reflection squares to the identity, e.g. satisfies but is neither nor . All says is that is its own inverse.
A student writes "." Where's the slip?
They added instead of subtracting the off-diagonal product. The determinant is ; the minus is what encodes signed area, so it can never be dropped.
A student says " exists, so has the same entries as just rearranged." True in general?
No. For most matrices the entries also get scaled by , which changes their numeric values, not just their positions. Only when (or ) is a pure rearrangement/sign-flip of 's entries.
Why questions
Why does dividing by appear in the formula rather than, say, dividing by ?
Because is precisely the common factor that popped out of solving all four defining equations . It is the single quantity every entry of the answer shares.
Why is the exact boundary between invertible and not, rather than something like ?
Because measures area, and an inverse exists precisely when no area is destroyed. Any nonzero area — however tiny — can be rescaled back; only zero area loses information irreversibly.
Why does a negative determinant still give a valid inverse, physically?
A negative det means the map flips orientation (turns the plane over like a page). Flipping is reversible — you just flip back — so orientation reversal never blocks an inverse; only area collapse does.
Why must we compute the determinant before applying the inverse recipe?
If it's zero, the recipe demands dividing by zero and no inverse exists — so checking first saves you from producing a meaningless answer with undefined entries.
Why is and not ?
Because undoing a sequence reverses its order. applies 's effect first when acting on a vector; to reverse, you must undo the last-applied thing first, i.e. undo last, so comes first in the product.
Why does the geometric picture (columns forming a parallelogram) guarantee that parallel columns mean no inverse?
Parallel columns span a degenerate parallelogram of zero area, so . Zero area means the whole plane is squashed onto a line — many inputs share one output — and a function that isn't one-to-one cannot be undone.
Edge cases
Is the zero matrix invertible?
No. Its determinant is , and it sends every vector to , erasing all information — the most extreme singular case possible.
If has one row entirely zero, what can you say?
It is singular. A zero row forces (e.g. gives ), so no inverse exists.
Does a diagonal matrix always have an inverse?
Only when both and . Then and ; if either diagonal entry is , that axis is collapsed and the matrix is singular.
What is the inverse of the identity-scaled matrix for ?
. Uniform scaling by is undone by uniform scaling by ; this fails only at , where everything collapses to the origin.
Can a matrix equal its own adjugate?
Yes — for example has adjugate (swap does nothing on equal diagonal, the zeros stay zero). This happens when and .
If , how does relate to the adjugate of ?
They are equal: . Dividing by changes nothing, so the inverse is just the swap/negate matrix with no rescaling.
Recall One-line summary of every trap here
An inverse exists (any nonzero value, positive or negative); the recipe is swap diagonal, negate off-diagonal, divide by det; and order reverses under . Everything else is a variation on forgetting one of these.
Connections
- Inverse of 2×2 matrix — parent note with the full derivation and formula.
- Determinant of a 2×2 matrix — the gatekeeper behind every trap above.
- Singular vs non-singular matrices — the invertible/not dividing line probed by the edge cases.
- Matrix multiplication — why order reverses and why entrywise reciprocals fail.
- Identity matrix — the that self-inverts and defines the inverse.
- Solving linear systems — the left-multiply application in "Spot the error".
- Adjugate and cofactors — the swap/negate object several questions test.
- Linear transformations and area — the geometric heart of the "why" questions.