Goal
Construct the signed 3x3 cofactor matrix of a finite row-major matrix from its nine 2x2 minors.
Cofactor Matrix
Construct the signed 3x3 cofactor matrix of a finite row-major matrix from its nine 2x2 minors.
Key facts
- What it does
- Construct the signed 3x3 cofactor matrix of a finite row-major matrix from its nine 2x2 minors.
- Formula
Cij = (-1)^(i+j) det(Mij), where Mij is the 2x2 matrix left after deleting row i and column j.- You enter
- A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33
- Worked example
- C(A) = [[-24, 20, -5], [18, -15, 4], [5, -4, 1]].
A clearer path to an answer
From your question to a useful result
This page keeps the calculation transparent: define the goal, enter the matching values, inspect the method, and decide what the result means in your situation.
Inputs
A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33
Method
Cij = (-1)^(i+j) det(Mij), where Mij is the 2x2 matrix left after deleting row i and column j.
Next step
Calculate, review the assumptions below, then compare a related tool when the decision needs more context.
Cofactor Matrix
Construct the signed 3x3 cofactor matrix of a finite row-major matrix from its nine 2x2 minors.
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Result
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Calculation map
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01
Inputs (9)
- A11 Ready
- A12 Ready
- A13 Ready
- A21 Ready
- +5 more inputs
02
Formula
Cij = (-1)^(i+j) det(Mij), where Mij is the 2x2 matrix left after deleting row i and column j.
Bounded, transparent calculation
03
Result
- Calculate to preview the result.
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Formula, assumptions, and example
Formula: Cij = (-1)^(i+j) det(Mij), where Mij is the 2x2 matrix left after deleting row i and column j.
Each cofactor is a signed 2x2 minor. The nine cofactors form a new 3x3 matrix, and first-row expansion of those cofactors also supplies the determinant used by the inverse identity.
- A is a finite real 3x3 matrix entered in fixed row-major order.
- All entries and every 2x2 minor are required to remain finite within the bounded arithmetic contract.
- The result is the cofactor matrix itself, not its transpose, adjugate, determinant-only value, or inverse.
Worked example: C(A) = [[-24, 20, -5], [18, -15, 4], [5, -4, 1]].
Displayed input contract
- A11 · minimum -1000 · maximum 1000
- A12 · minimum -1000 · maximum 1000
- A13 · minimum -1000 · maximum 1000
- A21 · minimum -1000 · maximum 1000
- A22 · minimum -1000 · maximum 1000
- A23 · minimum -1000 · maximum 1000
- A31 · minimum -1000 · maximum 1000
- A32 · minimum -1000 · maximum 1000
- A33 · minimum -1000 · maximum 1000
The displayed limits are checked before the handler runs. Model-specific domain checks may also reject impossible or non-finite inputs.
Methodology: This calculator follows the WorldCalculate input, formula, precision, and boundary policy. Read the official methodology.
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Answer-first guide
How to use the Cofactor Matrix for a real question
Construct the signed 3x3 cofactor matrix of a finite row-major matrix from its nine 2x2 minors. Start with one clearly defined goal, enter values in the units shown, and keep the result attached to the assumptions below.
What this answers
This tool is useful when your question includes cofactor matrix, 3x3 cofactors, signed minors. It returns the outputs declared in the calculator contract rather than a live quote, approval, diagnosis, or professional sign-off.
What you enter
A11 · A12 · A13 · A21 · A22 · A23 · A31 · A32 · A33. Keep the same time period, unit system, and currency wherever the form requires comparable values.
How to check it
Run the worked example first, compare its output with the page's example, then change one input at a time. This makes an unexpected result easier to trace to a unit, boundary, or assumption.
Three checks before you rely on the answer
- Match the question. Confirm that the result means the quantity you need, not a similar-sounding percentage, balance, rate, or estimate.
- Match the inputs. Use the requested units and period, and read each hint before replacing the example values with your own.
- Read the boundary. Review the assumptions and limits. A is a finite real 3x3 matrix entered in fixed row-major order.
Need a wider view? Browse Math Calculators or compare the related tools below. The WorldCalculate methodology explains how formulas, examples, limits, and revisions are reviewed.
How to use the Cofactor Matrix
- Enter A11 — Row 1, column 1 of matrix A. (matrix entry).
- Enter A12 — Row 1, column 2 of matrix A. (matrix entry).
- Enter A13 — Row 1, column 3 of matrix A. (matrix entry).
- Enter A21 — Row 2, column 1 of matrix A. (matrix entry).
- Enter A22 — Row 2, column 2 of matrix A. (matrix entry).
- Enter A23 — Row 2, column 3 of matrix A. (matrix entry).
- Enter A31 — Row 3, column 1 of matrix A. (matrix entry).
- Enter A32 — Row 3, column 2 of matrix A. (matrix entry).
- Enter A33 — Row 3, column 3 of matrix A. (matrix entry).
- Choose Calculate and read the result panel.
- Use Download PDF or Download Word to save a result sheet.
Formula
Cij = (-1)^(i+j) det(Mij), where Mij is the 2x2 matrix left after deleting row i and column j.
Each cofactor is a signed 2x2 minor. The nine cofactors form a new 3x3 matrix, and first-row expansion of those cofactors also supplies the determinant used by the inverse identity.
Worked example
C(A) = [[-24, 20, -5], [18, -15, 4], [5, -4, 1]].
Assumptions and limits
- A is a finite real 3x3 matrix entered in fixed row-major order.
- All entries and every 2x2 minor are required to remain finite within the bounded arithmetic contract.
- The result is the cofactor matrix itself, not its transpose, adjugate, determinant-only value, or inverse.
Who uses this calculator?
- Students learning minors and determinant expansion
- Developers testing an inverse-matrix support routine
- Analysts needing signed local minors of a fixed 3x3 matrix
When is it useful?
- Build the cofactor matrix for a 3x3 determinant exercise.
- Prepare the adjugate step of an inverse calculation.
- Check the checkerboard signs of nine signed minors.
Context and background
The mathematical structure behind the tools
Math calculators move from named quantities to a relation, then to a result that can be checked with substitution, units, or an alternate form.
Arithmetic, algebra, geometry, trigonometry, and number theory provide reusable structures for classroom work and everyday reasoning. Each page narrows that structure to one declared problem.
Research and review
How this guide was researched
Researched by Hassan ALRowaie, Founder and editorial researcher at WorldCalculate.
This guide follows the live calculator's declared inputs, formula, worked example, assumptions, validation boundaries, and source-backed methodology. The review date describes editorial review of the calculator explanation; it is not a promise that external facts or rates remain current.
A cofactor matrix records the signed 2x2 minors associated with every entry of a 3x3 matrix. This calculator accepts nine finite row-major entries, deletes one row and one column for each position, applies the checkerboard sign, and returns the nine cofactors as deterministic matrix text. It also computes the determinant internally for a transparent first-row expansion step, but it does not divide by that determinant and does not return an inverse. The guide explains minors, signs, field order, a complete worked example, zero and singular cases, finite arithmetic, the relationship to the adjugate and inverse, output formatting, and responsible use.
What a cofactor is
For position (i, j) in A, the minor determinant Mij is formed by deleting row i and column j, leaving a 2x2 matrix. The cofactor Cij is the determinant of that remaining matrix multiplied by (-1)^(i+j). The nine Cij values are placed into a new 3x3 matrix in their own row-major positions. This operation uses the source position twice: once to choose which row and column to delete, and once to choose the sign in the checkerboard pattern.
The cofactor matrix is not the same as the adjugate. The adjugate is the transpose of the cofactor matrix. That distinction is essential when using the identity A^-1 = adj(A)/det(A). This page stops at the cofactor matrix and shows the determinant expansion as a check. It does not silently transpose the result, because a caller who needs an adjugate should see and record that additional structural step.
- Each cofactor comes from one deleted-row, deleted-column 2x2 minor.
- The sign follows a checkerboard pattern.
- Cofactor matrix and adjugate differ by a transpose.
The fixed 3x3 input
The source matrix is [[a11, a12, a13], [a21, a22, a23], [a31, a32, a33]]. For C11, delete row one and column one, leaving entries a22, a23, a32, and a33. For C12, delete row one and column two, leaving the columns one and three of the bottom two rows. The other seven positions follow the same deletion rule. Writing the full grid first reduces the chance of selecting a neighboring value from the wrong row or column.
All nine source values are finite numbers in the inclusive range -1000 through 1000. The cofactor products and differences are finite-checked as they are formed. A string, omitted property, NaN, infinity, or out-of-range number is rejected even if a particular minor would not use the missing position in a careless implementation. The complete matrix contract makes every cofactor reproducible and keeps the output suitable for JSON serialization.
- Use a11 through a33 in row-major order.
- Every source entry belongs to several different minors.
- Finite validation covers source values and derived minors.
The checkerboard signs
For a 3x3 matrix, the cofactor signs are positive, negative, positive in the first row; negative, positive, negative in the second row; and positive, negative, positive in the third row. This is the pattern +, -, + / -, +, - / +, -, +. It comes from (-1)^(i+j). The underlying 2x2 minor can itself be negative, so the final sign is the product of the minor's arithmetic sign and the checkerboard sign. A visual sign pattern alone is not enough; calculate the minor first.
The alternating signs are what make determinant expansion work. Expanding along the first row gives det(A) = a11 C11 + a12 C12 + a13 C13. The engine computes all nine cofactors, then uses the first-row cofactors to calculate a finite determinant for the steps and note. This internal determinant is not an extra matrix result and does not change the returned cofactor matrix.
- Rows one and three start with a positive cofactor sign.
- Row two starts with a negative cofactor sign.
- The minor's own sign must be combined with the checkerboard sign.
A complete cofactor example
For A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], C11 is 1x0 - 4x6 = -24. C12 is -(0x0 - 4x5) = 20. C13 is 0x6 - 1x5 = -5. These three values form the first cofactor row. Continuing with the second source row gives C21=18, C22=-15, and C23=4. The third row gives C31=5, C32=-4, and C33=1.
The returned matrix is [[-24, 20, -5], [18, -15, 4], [5, -4, 1]]. The determinant check is 1x(-24) + 2x20 + 3x(-5) = 1. This example is useful because the determinant is simple while the cofactors exercise every checkerboard sign. A reviewer can independently recompute one minor from each row and then verify the first-row expansion.
- The first cofactor row is [-24, 20, -5].
- The second cofactor row is [18, -15, 4].
- The third cofactor row is [5, -4, 1].
Zeros and singular matrices
A zero source entry is valid and can simplify several minors. A singular matrix is also allowed for the cofactor operation because no division is performed. Its cofactor matrix can contain nonzero values even when the determinant is zero. For example, a matrix with two equal rows has determinant zero, but deleting one of those rows can leave a nonzero 2x2 minor. The calculator therefore does not reject a singular input merely because it cannot be inverted.
This behavior distinguishes cofactor construction from inversion. The inverse tool uses the same determinant and cofactor identity but must reject a determinant that is zero or too close to zero before dividing. Keeping the cofactor result available for singular matrices is mathematically useful and makes the boundary between a defined minor and an undefined inverse explicit.
- Zero entries are valid source values.
- Cofactors can be nonzero for a singular matrix.
- Singularity matters for inverse division, not for forming minors.
Finite arithmetic and matrix text
Each 2x2 minor uses two products and one subtraction. With source entries bounded by 1000, these values remain well inside ordinary finite arithmetic, but the implementation still checks every product, difference, cofactor, and determinant. This protects the result tree from hidden nonfinite values and makes any future boundary change fail clearly. Negative zero is normalized before values are formatted into the text matrix.
The matrix result is rendered as [[row one], [row two], [row three]] with comma separators and a fixed row-major order. The steps are strings and include the determinant expansion, while the cofactor result itself is text because it is a matrix. A caller should retain the source matrix and not attempt to recover it from the formatted cofactor string. JSON serialization is safe for the complete handler output.
- Products, minors, cofactors, and determinant are finite-checked.
- The signed matrix uses deterministic row-major text.
- Keep source entries with the formatted cofactor result.
Cofactors, adjugate, and inverse
The cofactor matrix supplies the next step in the classical inverse identity. Transpose it to obtain the adjugate, then divide every adjugate entry by det(A). The transpose is not optional: using the cofactor matrix without transposing generally places inverse entries in the wrong positions. This calculator returns only the cofactor matrix so that each stage can be inspected and tested independently. The inverse page repeats the identity and performs the guarded division.
Determinant expansion can be performed along any row or column when the matching cofactors are used, although the calculator displays the first-row expansion for a stable, concise audit. The determinant does not need to be recomputed from a separate parser. The page uses the fixed 3x3 formula directly, which makes the sign and minor dependencies visible and avoids accepting an arbitrary matrix expression from a user.
- Transpose the cofactor matrix to form the adjugate.
- Divide by the determinant only in an inverse contract.
- First-row expansion is a check, not an extra output matrix.
Verification and responsible use
To verify cofactors, write the sign checkerboard, compute the four entries of each 2x2 minor, and compare one complete row at a time. Use a diagonal matrix to check that off-diagonal cofactors follow the expected zero pattern, and use a matrix with repeated rows to confirm that singular inputs are still accepted. Recompute the determinant from the returned first-row cofactors and compare it with an independent expansion. These tests check deletion, signs, arithmetic, and result placement together.
The cofactor matrix is an algebraic building block, not a guarantee that an inverse is stable or that a matrix models a real system. The page does not choose precision, infer units, estimate uncertainty, or assess a downstream solve. Enter only numeric values, preserve the source matrix and assumptions, and apply qualified review when cofactors feed a high-consequence calculation. A correctly constructed cofactor matrix can still be unsuitable if the surrounding model is ill-conditioned or conceptually wrong.
- Recompute minors with the checkerboard signs visible.
- Use diagonal and repeated-row fixtures.
- Do not treat cofactor correctness as a stability or domain certificate.
The deletion map for every position
Each source position determines a unique pair of deleted indices. C11 deletes row one and column one, C12 deletes row one and column two, and C13 deletes row one and column three. The second and third source rows follow the same map. A reliable implementation can write the remaining four source keys beside each cofactor before multiplying them. This makes it possible to audit a result without relying on a compact closed-form expression whose indexing may be difficult to inspect.
The cofactor output keeps the position of the source label: C21 is placed in row two, column one of the returned cofactor matrix even though its minor was formed from rows one and three. This is not an adjugate layout. If a downstream step expects an adjugate, it must transpose the completed cofactor matrix deliberately. Keeping the two layouts distinct avoids a result that is numerically populated but structurally mislabeled.
- Every Cij has a defined deleted row and column.
- Minor source keys should be auditable independently.
- Cij stays at position ij until an explicit transpose forms the adjugate.
Two-by-two minor arithmetic
A 2x2 determinant [[p, q], [r, s]] is p x s - q x r. The order of the two products matters because swapping the terms changes the sign. For each cofactor, calculate the first diagonal product, calculate the second diagonal product, subtract, and only then apply the checkerboard sign. This sequence helps a reviewer distinguish a wrong minor selection from a wrong sign and makes small hand fixtures easy to reproduce.
With entries bounded by 1000, an individual product is bounded by one million before subtraction. The handler still finite-checks every product and difference because the arithmetic contract should fail clearly if the bounds or formula are changed later. A negative minor is valid. Applying a negative checkerboard sign to it can produce a positive cofactor, so the final sign cannot be inferred from the sign pattern alone.
- A 2x2 determinant is one product minus another.
- Calculate the minor before applying its checkerboard sign.
- Negative minors and positive cofactors can occur together.
Checkerboard sign verification
The sign at position ij is positive when i+j is even and negative when i+j is odd. For a 3x3 result this produces positive corners and center, with negative edge positions that have one index even and the other odd. Write the signs over the output grid before checking values. A test that asserts only determinant magnitude can miss a sign error that changes the cofactor matrix while accidentally preserving one expansion under a particular input.
Use a matrix with distinct signed entries so every position exercises a different minor and sign. A diagonal matrix is useful for a structural pattern, but its zeros can conceal a misplaced term. A second fixture with repeated or zero values can then check simplification. Together these cases expose both the general deletion map and the branches where arithmetic happens to cancel.
- The sign is determined by the sum of the indices.
- Distinct signed fixtures expose checkerboard mistakes.
- Diagonal fixtures should supplement, not replace, general cases.
Determinant expansion as a cross-check
The first-row expansion det(A) = a11 C11 + a12 C12 + a13 C13 links three returned cofactors to the determinant. The cofactor handler uses this relation for its steps, so a reviewer can recompute a scalar without implementing all nine additional minors. Expansion along a different row or column should agree when the matching cofactors are used. That independent route is valuable when debugging a single misplaced cofactor.
The determinant is an internal check here, not the requested result. It must still be finite-checked because it is derived from bounded cofactors and source values. The handler does not expose a determinant result object, and it does not reject a zero determinant. Those choices preserve the distinction between constructing signed minors and deciding whether division for an inverse is stable.
- First-row expansion connects cofactors to determinant arithmetic.
- A different expansion can independently check placement.
- The internal determinant does not change the cofactor output contract.
Cofactors of structured matrices
For a diagonal matrix, diagonal cofactors are products of the other two diagonal entries, while many off-diagonal cofactors are zero. This provides a predictable pattern but should be checked against the deletion rule rather than assumed. For a triangular matrix, some minors simplify but the checkerboard signs remain active. For a zero matrix, every minor and cofactor is zero. These structures give compact regression fixtures for source selection, signs, and zero normalization.
A repeated-row matrix is singular, yet its cofactors can carry information about the two-by-two submatrices left after deleting a row. This is why the cofactor operation accepts singular inputs. Compare its determinant expansion with zero and inspect whether nonzero cofactors remain. The result should not be described as an inverse or as evidence that the rows are independent.
- Diagonal and triangular forms provide structural patterns.
- The zero matrix produces a zero cofactor matrix.
- Singular inputs can have useful nonzero cofactors.
Cofactor matrix and adjugate workflow
The adjugate is C(A)^T, so the off-diagonal cofactor entries exchange positions before inverse division. A caller that stores a cofactor matrix should label it explicitly and should not call it an adjugate until that transpose is performed. This matters even when the source matrix is not symmetric. A symmetric source can produce a symmetric cofactor matrix and hide an omitted transpose, so include a nonsymmetric fixture in integration checks.
The inverse identity adds another condition: det(A) must be nonzero and sufficiently separated from zero under the inverse page's scale-relative policy. Cofactor construction has no such division boundary. Keeping the stages separate lets a workflow inspect minors for a singular matrix without pretending that the next inverse stage is available.
- Adjugate equals the transpose of the cofactor matrix.
- Nonsymmetric fixtures reveal an omitted adjugate transpose.
- Inverse stability is a later contract, not a cofactor precondition.
Signs, cancellation, and finite bounds
A cofactor can be close to zero because the two products in its minor nearly cancel. That does not mean either source product was small. Preserve the numeric result and the source entries when a near-zero cofactor matters. The formatter normalizes negative zero for stable JSON-safe text, but it does not apply an arbitrary small-value threshold. A caller that treats small cofactors as zero must state its own tolerance and keep it separate from this arithmetic result.
The source bounds and finite checks protect the browser-facing result, but they do not certify a condition number or a physical interpretation. If an application needs larger values, exact integers, interval bounds, or uncertainty propagation, it needs a revised contract and tests. Do not silently coerce strings or clip an out-of-range entry because doing so changes the minor and every related cofactor.
- Near-zero cofactors can result from cancellation.
- Negative zero is normalized without thresholding small nonzero values.
- Bounds protect the page but do not certify conditioning.
Cross-tool verification
A focused verification can calculate cofactors from a separate minor table, compare all nine positions, and then use the first-row expansion to check the determinant. Transpose the returned cofactor matrix and compare it with the adjugate used by the inverse handler for the same nonsingular fixture. Finally multiply the source matrix by the adjugate and confirm the result equals det(A) times I3 within a declared tolerance. These checks cover placement before division and identity behavior after it.
Do not use only the default matrix for cross-tool evidence. Add a diagonal matrix, a zero matrix, a repeated-row singular matrix, a nonsymmetric signed matrix, and a matrix with large allowed entries. The singular and zero cases are especially important because they prove the cofactor page remains defined where the inverse page rejects. Keep the input object immutable through every call.
- Compare all nine cofactors with an independent minor table.
- Use adjugate and determinant identity checks for nonsingular fixtures.
- Include singular cases that inverse calculation must reject.
Provenance and downstream meaning
A reproducible cofactor record keeps the full named source matrix, the row-major convention, the returned cofactor matrix, the internal determinant check, and the operation name. If the source came from a larger table, retain the mapping from labels to a11 through a33. The calculator has no field for units, timestamps, or private identifiers, so those details belong in a controlled surrounding record rather than being inserted into numeric inputs or explanatory strings.
Cofactors can appear in symbolic derivations, inverse construction, determinant sensitivity, and software tests. The same nine-number output can mean different things in those workflows. Do not infer that a nonzero cofactor proves invertibility, that a zero cofactor proves dependence, or that the matrix represents a safe real system. Interpret the result only after identifying the downstream identity and its assumptions.
- Preserve source mapping and the determinant check.
- Cofactor values have workflow-specific meanings.
- Do not infer invertibility or safety from one cofactor pattern.
Final cofactor acceptance criteria
A cofactor result is ready for handoff when each of the nine finite source entries is mapped correctly, each signed 2x2 minor uses the correct deleted row and column, every arithmetic intermediate is finite, and the result is placed at the matching Cij position. The output should use stable row-major punctuation, include a determinant expansion step, and retain the complete source matrix. A visual check of one corner is not enough because the center and edge signs follow different checkerboard positions.
The remaining questions concern the next algebraic or domain operation. Whether to transpose into an adjugate, divide by a stable determinant, solve a system, or publish an interpretation must be decided by a separate reviewed contract. This page should remain a transparent, bounded cofactor constructor that is useful for education, testing, and controlled matrix workflows without overstating what its output proves.
- Confirm deletion, sign, arithmetic, and output position for all nine values.
- Retain source matrix and determinant check with the result.
- Separate cofactor construction from inverse and domain conclusions.
Frequently asked questions
What is the Cofactor Matrix?
Construct the signed 3x3 cofactor matrix of a finite row-major matrix from its nine 2x2 minors.
What is the formula for the Cofactor Matrix?
Cij = (-1)^(i+j) det(Mij), where Mij is the 2x2 matrix left after deleting row i and column j. Each cofactor is a signed 2x2 minor. The nine cofactors form a new 3x3 matrix, and first-row expansion of those cofactors also supplies the determinant used by the inverse identity.
What do I need to use this calculator?
Enter A11, A12, A13, A21, A22, A23, A31, A32, A33, then choose Calculate.
What are the limits of this calculator?
A is a finite real 3x3 matrix entered in fixed row-major order. All entries and every 2x2 minor are required to remain finite within the bounded arithmetic contract. The result is the cofactor matrix itself, not its transpose, adjugate, determinant-only value, or inverse.
Methodology
This calculator is part of the WorldCalculate library. Its formula, example, assumptions, input bounds, and output formatting follow the official methodology.
Use this calculator as part of a bigger plan
These WorldCalculate collections connect this tool with related questions while keeping each calculation separate and transparent.