Determinant by Expansion Calculator

Enter Matrix Values

Choose a square matrix size, select a row or column for cofactor expansion, then enter values. Empty cells are treated as zero.

Matrix Size

Expansion Mode

Line Number

Display Precision

Matrix Input Grid

Formula Used

Laplace expansion computes the determinant by selecting one row or one column, then summing each entry multiplied by its cofactor.

General rule: det(A) = Σ a_ijC_ij

Cofactor rule: C_ij = (-1)^i+j × det(M_ij)

Here, M_ij is the minor matrix formed after removing row i and column j. The calculator evaluates each minor recursively until smaller determinants are reached.

How to Use This Calculator

Select a square matrix size from 2×2 through 5×5.
Choose auto, row, or column expansion mode.
Pick the line number when using manual expansion.
Enter each matrix value in the responsive grid.
Click the calculate button to show the determinant above the form.
Review minor matrices, cofactors, and term contributions.
Export the summary using the CSV or PDF buttons.

Example Data Table

This example uses a 3×3 matrix. Expanding across row 2 gives the same determinant as any valid row or column choice.

Matrix Size	Expansion Choice	Matrix Entries	Expected Determinant
3 × 3	Row 2	[2, 1, 3] [0, -1, 4] [5, 2, 1]	17

Frequently Asked Questions

1. What does determinant by expansion mean?

It means finding the determinant through Laplace expansion. You choose one row or column, build minors, apply alternating signs, compute cofactors, and add all term contributions.

2. Why does auto mode choose rows or columns with zeros?

Zeros remove entire terms from the expansion. That reduces minor calculations and makes the determinant faster to compute and easier to verify by hand.

3. Can this calculator handle decimal entries?

Yes. You can enter integers, decimals, negative values, and zeros. The result and each intermediate term are displayed using your selected output precision.

4. Is the determinant the same from every expansion line?

Yes. Any valid row or column expansion should produce the same determinant. Different choices only change the amount of arithmetic, not the final answer.

5. Why are cofactors signed with plus and minus patterns?

Cofactor signs follow the checkerboard pattern from (-1)^i+j. This preserves the correct orientation of the determinant and ensures the expansion stays mathematically valid.

6. What matrix sizes are supported here?

This version supports square matrices from 2×2 up to 5×5. That range is practical for detailed expansion steps without making the output too large.

7. What does the minor determinant represent?

It is the determinant of the smaller matrix left after removing one row and one column. Each minor helps build the cofactor for an expansion term.

8. When should I use expansion instead of row reduction?

Expansion is ideal for learning, proof work, symbolic patterns, and sparse matrices. Row reduction is often quicker for larger dense matrices when steps are not required.