AlgebraFree

Determinant Calculator with Steps

Compute a 3x3 determinant with cofactor expansion details shown in the results.

Enter Values

a11

a12

a13

a21

a22

a23

a31

a32

a33

Result

Enter values above and click Calculate to see your result.

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Formula

det = a(ei - fh) - b(di - fg) + c(dh - eg)

First-row cofactor expansion with 2x2 minors.

Worked Example

Enter nine matrix entries and the tool lists minors and the final determinant.

Step-by-Step Cofactor Expansion for 3x3 Determinants

This calculator breaks down the 3x3 determinant computation into visible steps so you can follow the cofactor expansion process and verify each intermediate calculation.

Step 1: For each element in the first row, form the 2x2 minor by deleting that element row and column
Step 2: Compute each 2x2 minor determinant using the ad - bc formula
Step 3: Apply the alternating sign pattern (+, -, +) to each term
Step 4: Multiply each first-row element by its signed minor and sum all three terms
The result shows each minor value, the signed contribution, and the final determinant sum

Following these steps by hand is the standard method taught in linear algebra courses. This calculator verifies your work and catches arithmetic errors in the intermediate steps.

You can also calculate changes using our 3x3 Determinant Calculator, Determinant Calculator, Matrix Determinant Calculator or 2x2 Determinant Calculator.

Frequently Asked Questions

How do I read the step-by-step output?

Each step shows one term from the first-row expansion: the element value, its 2x2 minor determinant, the sign (+/-), and the contribution to the total. The final row sums all three contributions to give the determinant.

Can I use this for 4x4 or 2x2 matrices?

This page shows steps for 3x3 only. The 2x2 formula (ad - bc) needs no step breakdown. For 4x4, use the general determinant calculator, though step details are less practical due to the number of terms.

Why do the signs alternate +, -, +?

The cofactor sign for position (i,j) is (-1) raised to (i+j). For the first row (i=1): positions j=1,2,3 give signs (+1), (-1), (+1). This checkerboard pattern ensures the expansion produces the correct determinant.

Can I expand along a different row or column?

Yes, in theory. Expanding along any row or column gives the same determinant. This calculator uses the first row, which is the most common choice for teaching and manual computation.

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