Inverse Matrix Calculator
The 2×2 inverse by the adjugate over the determinant — fraction form kept, decimal form beside it, and the proof printed: A × A⁻¹ recomputed live to the identity.
Inverse Matrix Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- 2×2 only — 3×3 needs elimination (out of scope)
- Exact fraction portrait shown for integer entries; decimal entries get the decimal portrait
In short: [4 7; 2 6]: det = 4×6 − 7×2 = 10, so the inverse is (1/10)·[6 −7; −2 4] = [0.6 −0.7; −0.2 0.4]. The proof card multiplies A by its inverse and prints what comes out: [1 0; 0 1] — the identity, recomputed live, not asserted. Det = 0 is refused with the honest reason: no matrix multiplies back to I.
Formula
[a b; c d]⁻¹ = (1/det)·[d −b; −c a]
swap the diagonal, negate the off-diagonal, divide by det
the swapped-and-negated grid is the adjugate; the division is why det = 0 kills the inverse.
Worked Example
- Determinant first. ad − bc — if it lands on zero the page stops with the honest verdict, because the rule would divide by zero.
- Adjugate. swap a and d, negate b and c — no arithmetic beyond signs yet.
- Divide and prove. multiply every entry by 1/det, then recompute A × A⁻¹ live: the four cells must read 1, 0, 0, 1.
The fraction form is the exact answer — (1/10)·[6 −7; −2 4] — and the decimal form is its portrait. The proof card is the part worth trusting: it is computed from YOUR matrix and the YOUR-inverse product, so if the four cells ever failed to read 1, 0, 0, 1 the page would be lying in public.
Strengths & Limits Of This Model
Where this engine is strong
- The identity product is recomputed live — the proof ships with the answer
- Singular refusal comes with the geometry, not just a message
Where it stops
- No 3×3 inverses
- No Gaussian elimination path
Practical Use Cases
Solving Ax = b
x = A⁻¹b — one multiply
Undoing transforms
the matrix that un-rotates, un-scales, un-shears
Homework
the adjugate law with its division kept visible
Methodology & Editorial Standards
Matrix parsed and validated 2×2. det = ad − bc; zero is refused with the geometric verdict. Inverse = (1/det)·[d −b; −c a], printed in exact fraction form (1/det times the integer adjugate when the entries are integers) and in decimal portrait. The proof card recomputes A·A⁻¹ and prints the four cells, which must read 1, 0, 0, 1.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Inverse Matrix Calculator — 9 Expert FAQs
9 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What is an inverse matrix trying to be?
The undo button. A⁻¹ is the matrix with A·A⁻¹ = I — multiply by A then by A⁻¹ and nothing has happened, exactly like 7 and 1/7 for numbers. The proof card prints that product recomputed live, because an inverse without its identity check is a rumor.
Why does det = 0 mean no inverse exists?
Because the adjugate rule divides by the determinant, and division by zero is not a value — but the geometry is the real reason: det = 0 means the map crushed the plane onto a line, and no function can un-crush it (two different points now share one image; the undo cannot know which to return). Singular is permanent.
What is the adjugate, concretely?
The minor surgery: swap the two diagonal entries, negate the two off-diagonal ones. [a b; c d] becomes [d −b; −c a] before any division happens — which is why the fraction card can show (1/10)·[6 −7; −2 4] exactly, with the determinant visible as the scale of the whole grid.
Why keep the fraction form at all?
Because it is EXACT. [0.6 −0.7; −0.2 0.4] is the same matrix in decimal dress, but (1/10)·[6 −7; −2 4] shows the determinant as part of the structure — and for dets like 3 the decimal form repeats forever while 1/3 stays one symbol. Exactness first, portrait second.
How do I use an inverse to solve equations?
For Ax = b: x = A⁻¹b. The inverse undoes the mapping A applied to your unknown vector, leaving x bare. That is also why singular matrices block solving: with no undo available, Ax = b has either no solution or infinitely many — the same dichotomy the system solver names.
What does the inverse of a diagonal matrix look like?
Diagonal again, with reciprocals: [2 0; 0 3]⁻¹ = [0.5 0; 0 1/3…]. Each axis is scaled independently and each undo is its own reciprocal. Drive the chip and watch the proof card confirm — the pattern makes the general law memorable.
Does the page handle 3×3 inverses?
No — 2×2 only, and the limit is stated. The adjugate at 3×3 is nine cofactors needing nine 2×2 determinants: real work, done by elimination in practice, and a different page if this site ever grows one. The 2×2 case is where the LAW is visible in one glance.
Why does the proof card matter more than the answer?
Because it is the only part that cannot lie quietly. The inverse cells are arithmetic; the product A·A⁻¹ is a CHECK — four cells that must read exactly 1, 0, 0, 1. If a hand-copied inverse fails that product, the error is found in seconds; if it passes, the answer is beyond doubt.
Can decimals be entries?
Yes — any finite numbers. The determinant may come out decimal (then the fraction card shows 1/det with det as computed), and all printed cells carry the site’s six-place honesty. Singular detection is exact at det = 0 in double arithmetic.