Cofactor Expansion Calculator — Laplace Expansion

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Walk-through

How to Use This Calculator

4 steps

1

Pick a matrix size

Choose 3×3 or 4×4 from the size selector. The entry grid resizes to match, and a worked example loads automatically so you can see the method in action right away.

2

Enter your matrix values

Click any cell in matrix A and type a number. The determinant, minors, cofactors, and expansion steps all update instantly as you edit — there's no Calculate button to press.

3

Choose a row or column to expand along

Use the Expand Along selector to pick any row or column. The determinant always comes out the same no matter which one you choose — picking the one with the most zeros just makes the arithmetic shorter.

4

Read the minors, cofactors, and steps

The Minors & Cofactors tab shows the checkerboard sign pattern plus every minor matrix and its determinant for the row or column you chose. The Expansion Steps tab shows the full substituted equation from the formula down to the final answer.

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Reference

Formula & Methodology

2 formulas

Laplace (cofactor) expansion

det(A) = Σⱼ (−1)^(i+j) · A[i][j] · Mᵢⱼ

To expand along row i, sum over every column j: multiply each entry A[i][j] by its cofactor, which is the signed determinant of the minor Mᵢⱼ — the (n−1)×(n−1) matrix left after deleting row i and column j. The sign alternates in a checkerboard pattern, (−1)^(i+j). Expanding along a column instead works the same way, summing over rows i for a fixed column j. The result is identical no matter which row or column is chosen.

4×4 recursion

M₄ₓ₄ minor → expanded again down to a 2×2 base case

For a 4×4 matrix, each minor is itself a 3×3 matrix. This calculator computes that 3×3 minor's determinant directly (recursing internally down to the 2×2 base case ad − bc), so a 4×4 expansion always resolves to four fully worked 3×3 minors.

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Glossary

Key Terms Explained

7 terms

Cofactor The signed minor determinant at a given row and column: Cᵢⱼ = (−1)^(i+j) · Mᵢⱼ. The sign alternates in a checkerboard pattern starting with + at the top-left.

Minor The determinant of the smaller matrix left over after deleting one row and one column from the original matrix. A minor of a 3×3 matrix is a 2×2 determinant; a minor of a 4×4 matrix is a 3×3 determinant.

Laplace expansion Another name for cofactor expansion — the method of computing a determinant by summing signed products of entries and their cofactors along any single row or column.

Checkerboard sign pattern The alternating +/− pattern of (−1)^(i+j) across every cell of a matrix, starting with + in the top-left corner. It determines whether each term in the expansion is added or subtracted.

Determinant (det(A)) A single number computed from a square matrix that encodes properties like invertibility and volume scaling. Cofactor expansion is one general method for computing it.

Expansion row/column The row or column chosen as the basis for the expansion. Any row or column gives the same final determinant — choosing one with more zero entries reduces the number of minors that need to be computed.

Cofactor matrix The matrix formed by replacing every entry with its own cofactor. It's an intermediate step used by other methods (like the adjugate/inverse formula) but this calculator only computes the cofactors needed for the row or column you've chosen.

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Scenarios

Real-World Examples

3 worked examples

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3×3 expanded along row 1

A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]

Matrix A [[1, 2, 3], [0, 1, 4], [5, 6, 0]] Expand along Row 1

Expanding along row 1: det(A) = 1·M₁₁ − 2·M₁₂ + 3·M₁₃ = 1·(1·0 − 4·6) − 2·(0·0 − 4·5) + 3·(0·6 − 1·5) = 1·(−24) − 2·(−20) + 3·(−5) = −24 + 40 − 15 = 1. A classic textbook example — det(A) = 1 no matter which row or column you expand along.

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4×4 along a zero-heavy row

A = [[2, 0, 1, 3], [0, 0, 5, 0], [1, 4, 2, 1], [3, 1, 0, 2]]

Matrix A [[2, 0, 1, 3], [0, 0, 5, 0], [1, 4, 2, 1], [3, 1, 0, 2]] Expand along Row 2

Row 2 has three zero entries, so three of the four cofactor terms vanish automatically — only the single 5 in that row needs its 3×3 minor computed. That's the efficiency payoff of choosing the zero-heaviest row or column: the same determinant with far less arithmetic than expanding along a row full of nonzero numbers.

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Reading the checkerboard sign pattern

Any n×n matrix

Position (1,1) + Position (1,2) −

Every position (i, j) gets a sign of (−1)^(i+j): + when the row and column indices sum to an even number, − when they sum to an odd number. Reading across any row (or down any column), the signs always alternate +, −, +, −, which is why it's called a checkerboard pattern. The Minors & Cofactors tab draws this full grid and highlights the row or column currently being expanded.

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Deep Dive

Understanding the Cofactor Expansion Calculator

4 chapters · 4 min read

Cofactor (Laplace) expansion is the general-purpose method for computing a matrix determinant by hand: pick any row or column, multiply each entry by its signed minor determinant, and add up the results. This calculator lets you pick that row or column yourself for a 3×3 or 4×4 matrix, then shows every minor, every cofactor, and the fully substituted equation — the same worked steps a linear algebra course would expect you to write out.

How cofactor expansion works

Choose a row (or column) to expand along. For each entry in that row, delete its row and column from the matrix to get a smaller "minor" matrix, then take that minor's determinant. Multiply the entry by its minor's determinant and by an alternating sign (+, −, +, − in a checkerboard pattern based on position), then add all of those signed products together. The result is the determinant of the whole matrix — and it comes out identical no matter which row or column you pick.

For a 4×4 matrix, each minor is itself a 3×3 matrix, so the calculator computes that smaller determinant directly before combining it into the top-level sum. The Expansion Steps tab shows the formula, the substituted numbers, and the running total that adds up to the final answer.

Inputs and what they mean

Choose a matrix size (3×3 or 4×4), fill in the grid with any real numbers, then pick a row or column from the Expand Along selector. There's no unit conversion involved — the calculator works on the numbers exactly as entered. Use the preset buttons to load the identity matrix, a worked example, or a blank grid to start from scratch.

Why the row or column choice matters (and doesn't)

Mathematically, every row and column gives the exact same determinant — that's a core property of cofactor expansion. Practically, though, some choices are much faster to compute by hand: a row or column with more zero entries means more of the cofactor terms vanish automatically, since anything multiplied by zero contributes nothing to the sum. Try switching the Expand Along selector between a zero-heavy row and a row full of nonzero numbers to see the arithmetic shrink.

Limits and edge cases

This calculator supports 3×3 and 4×4 matrices only. For a determinant computed with one fixed method plus a singular/invertible check, see the Matrix Determinant Calculator; for the inverse via the adjugate (cofactor matrix) method, see the Matrix Inverse Calculator; for larger matrices or Gauss-Jordan row reduction, use the general Matrix Calculator.

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Questions

Frequently Asked Questions

6 questions

What is the cofactor expansion formula?+

det(A) = Σⱼ (−1)^(i+j) · A[i][j] · Mᵢⱼ, where the sum runs over the entries of a chosen row i (or, symmetrically, over a chosen column j), and Mᵢⱼ is the determinant of the minor left after deleting that row and column.

What is a minor?+

A minor Mᵢⱼ is the determinant of the smaller matrix left over after deleting row i and column j from the original matrix. For a 3×3 matrix, every minor is a 2×2 determinant; for a 4×4 matrix, every minor is a 3×3 determinant.

What determines the +/− sign pattern?+

The sign is (−1)^(i+j), which produces a checkerboard pattern: + when the row and column indices sum to an even number, − when they sum to an odd number. The pattern always starts with + in the top-left corner.

Which row or column is best to expand along?+

Any row or column gives the same final determinant, but the one with the most zero entries takes the least arithmetic — every zero entry makes its whole term vanish without needing to compute a minor for it.

Do I get the same determinant no matter which row or column I choose?+

Yes, always. That's a fundamental property of the determinant — cofactor expansion along any row or any column of the same matrix produces the identical numeric result.

How does this work for a 4×4 matrix?+

Each entry's minor for a 4×4 matrix is a 3×3 matrix, so the calculator computes that 3×3 minor's determinant (recursing down to the 2×2 base case internally) before combining the four top-level terms into the final 4×4 determinant.

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Determinant