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Walk-through

How to Use This Calculator

5 steps

1

Pick a matrix size

Choose 2×2, 3×3, or 4×4 from the size selector on the Determinant tab. The entry grid rebuilds instantly to match, and any values that still fit the new size are kept.

2

Enter the matrix values

Type a number into each cell of the grid, row by row. Negative numbers and decimals are both fine — the determinant recalculates on every keystroke, no Calculate button required.

3

Read the determinant

The result card shows det(A) as soon as every cell is filled in, along with a plain-language line on whether the matrix is invertible or singular.

4

Open the Steps tab

Switch to Steps to see the full cofactor-expansion breakdown — for a 2×2 it's the direct ad − bc arithmetic; for 3×3 and 4×4 it's each signed term of the expansion along row 1, with its minor matrix shown.

5

Check invertibility, or try a preset

The Singular Check tab restates the same determinant against the det = 0 rule. Use the preset chips (2×2, 3×3, a singular 3×3, 4×4, and an identity matrix) to see each case worked out without typing anything.

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Reference

Formula & Methodology

3 formulas

Cofactor (Laplace) expansion

det(A) = Σ (−1)^(1+j) · a₁ⱼ · det(M₁ⱼ), for j = 1 to n

The determinant of an n×n matrix is computed by expanding along a row (here, row 1): multiply each entry in that row by the determinant of its minor — the smaller matrix left after deleting that entry's row and column — and add the results with alternating signs. This calculator applies the rule recursively, so the same formula covers 2×2, 3×3, and 4×4 matrices.

2×2 special case

det([[a, b], [c, d]]) = ad − bc

For a 2×2 matrix, cofactor expansion collapses to this single subtraction — it's the base case every larger determinant eventually expands down to.

Sign pattern

sign(i, j) = (−1)^(i+j)

Cofactor signs alternate in a checkerboard pattern starting with + in the top-left corner: +, −, +, − across row 1. This calculator always expands along row 1, so the signs used are +, −, +, − (up to 4 columns).

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Glossary

Key Terms Explained

7 terms

Determinant A single number computed from a square matrix that summarizes how the matrix scales area (2×2), volume (3×3), or hyper-volume (4×4). Written det(A) or |A|. A determinant of zero means the matrix collapses space into a lower dimension.

Cofactor A signed minor: the determinant of the matrix left after deleting one row and one column, multiplied by (−1) raised to the sum of that row and column's position. Cofactor expansion sums an entry times its cofactor across a row or column.

Minor The smaller (n−1)×(n−1) matrix left over after deleting one row and one column from an n×n matrix. Every cofactor is built from a minor's own determinant.

Singular matrix A square matrix whose determinant is exactly 0. Singular matrices have no inverse, and the linear system they represent either has no solution or infinitely many.

Invertible matrix A square matrix with a nonzero determinant. Invertible matrices have a unique inverse A⁻¹ such that A · A⁻¹ equals the identity matrix.

Laplace expansion Another name for cofactor expansion — the general method (credited to Pierre-Simon Laplace) for computing a determinant of any size by expanding along a chosen row or column.

Matrix A rectangular grid of numbers. This calculator only handles square matrices (equal rows and columns), since determinants are only defined for square matrices.

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Scenarios

Real-World Examples

3 worked examples

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2×2 by hand

A = [[4, 3], [6, 3]]

Size 2×2 Matrix [4, 3, 6, 3]

det(A) = ad − bc = (4)(3) − (3)(6) = 12 − 18 = −6. The determinant is nonzero, so this matrix is invertible — its rows point in genuinely different directions rather than collapsing onto the same line.

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3×3 cofactor walk

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

Size 3×3 Matrix [1, 2, 3, 0, 1, 4, 5, 6, 0]

Expanding along row 1: +1·det([1,4;6,0]) − 2·det([0,4;5,0]) + 3·det([0,1;5,6]) = 1(0−24) − 2(0−20) + 3(0−5) = −24 + 40 − 15 = 1. det(A) = 1, so the matrix is invertible.

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Singular matrix

A = [[1, 2, 3], [2, 4, 6], [3, 5, 7]]

Size 3×3 Matrix [1, 2, 3, 2, 4, 6, 3, 5, 7]

Row 2 is exactly twice row 1 — the rows are linearly dependent, so the determinant works out to exactly 0. This matrix is singular: it has no inverse, and a linear system built from it either has no solution or infinitely many.

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Deep Dive

Understanding the Matrix Determinant Calculator

3 chapters · 4 min read

A determinant condenses an entire square matrix down to a single number, and that number answers one of the most useful questions in linear algebra: does this matrix have an inverse? This calculator walks through the cofactor-expansion method used to compute a determinant by hand for a 2×2, 3×3, or 4×4 matrix, showing every signed term along the way.

How cofactor expansion works

Cofactor (Laplace) expansion computes a determinant by picking a row — this calculator always uses row 1 — and, for each entry in that row, multiplying it by the determinant of its minor (the matrix left after deleting that entry's row and column), with alternating signs starting at +. For a 2×2 matrix the process collapses to the single formula ad − bc. For 3×3, each of the three entries in row 1 is multiplied by the determinant of its 2×2 minor. For 4×4, each of the four entries is multiplied by the determinant of its 3×3 minor, which the calculator computes directly. Summing the signed terms gives the final determinant.

What a zero determinant means

A determinant of exactly 0 means the matrix is singular: its rows (or columns) are linearly dependent — one row can be written as a combination of the others, so the matrix squashes space into a lower dimension instead of stretching it into a full n-dimensional shape. Singular matrices have no inverse, and if the matrix represents a system of linear equations, that system either has no solution at all or has infinitely many. Any nonzero determinant, by contrast, means the matrix is invertible — a unique inverse A⁻¹ exists such that A · A⁻¹ gives the identity matrix.

Reading the sign pattern

The alternating + and − signs in cofactor expansion follow a checkerboard pattern: the sign of the cofactor for row i, column j is (−1) raised to the power (i + j). Since this calculator always expands along row 1 (i = 1), the signs simply alternate +, −, +, − across the columns. That pattern is why the middle term of a 3×3 expansion is always subtracted while the first and third are added — it isn't arbitrary, it falls directly out of the (−1)^(i+j) rule.

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Questions

Frequently Asked Questions

6 questions

How do you find the determinant of a 2×2 matrix?+

For A = [[a, b], [c, d]], det(A) = ad − bc — multiply the two diagonal entries together, then subtract the product of the two off-diagonal entries.

How do you find the determinant of a 3×3 matrix?+

Expand along a row using cofactor (Laplace) expansion: multiply each entry in the row by the determinant of the 2×2 minor left after deleting that entry's row and column, alternate the signs (+, −, +), and add the three results together.

What does det(A) = 0 mean?+

The matrix is singular — its rows or columns are linearly dependent. A singular matrix has no inverse, and any linear system it represents has either no solution or infinitely many.

What does it mean for a matrix to be invertible?+

A matrix is invertible exactly when its determinant is nonzero. Invertible matrices have a unique inverse A⁻¹ such that multiplying A by A⁻¹ produces the identity matrix.

What's the sign rule in cofactor expansion?+

Signs alternate in a checkerboard pattern: the cofactor for position (row i, column j) carries sign (−1)^(i+j). Expanding along row 1 gives the familiar +, −, +, − sequence across the columns.

Does this calculator show the steps?+

Yes — the Steps tab shows the full cofactor-expansion breakdown for whatever matrix you entered on the Determinant tab, including each signed term and its minor matrix for 3×3 and 4×4 sizes.

Download a one-page PDF of your numbers instantly. Add your email to also get our occasional calculator tips — no spam, unsubscribe anytime. Privacy.

Determinant	Result
det(A) ≠ 0	Invertible — the matrix has an inverse
det(A) = 0	Singular — no inverse exists

Matrix Determinant Calculator

Matrix

det(A)

Cofactor expansion

det(A)

Invertible or singular?

This matrix is

How to Use This Calculator

Pick a matrix size

Enter the matrix values

Read the determinant

Open the Steps tab

Check invertibility, or try a preset

Formula & Methodology

Key Terms Explained

Real-World Examples

2×2 by hand

3×3 cofactor walk

Singular matrix

Understanding the Matrix Determinant Calculator

How cofactor expansion works

What a zero determinant means

Reading the sign pattern

Frequently Asked Questions

Get a branded PDF of your results

Matrix Determinant Calculator

Matrix

det(A)

Cofactor expansion

det(A)

Invertible or singular?

This matrix is

How to Use This Calculator

Pick a matrix size

Enter the matrix values

Read the determinant

Open the Steps tab

Check invertibility, or try a preset

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Matrix Determinant Calculator

How cofactor expansion works

What a zero determinant means

Reading the sign pattern

Frequently Asked Questions

Get a branded PDF of your results

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