Determinant Using Cofactors Calculator

Calculator Inputs

Choose the matrix size, enter values, then expand across the first row using cofactors.

Matrix Size

Large screens: 3 columns Small screens: 2 columns Mobile: 1 column

a11

a12

a13

a21

a22

a23

a31

a32

a33

Example Data Table

Matrix Size	Matrix Entries	Expansion Row	Determinant
3 x 3	[2, 1, 3; 0, -1, 4; 5, 2, 1]	First row	31
4 x 4	[1, 2, 0, 3; 4, 1, -2, 5; 2, 0, 1, 4; 3, -1, 2, 1]	First row	62
2 x 2	[7, 2; 3, 9]	First row	57

Formula Used

For a square matrix A, the determinant by cofactors is:

det(A) = Σ a_1j C_1j, for j = 1 to n

Each cofactor equals C_1j = (-1)^1+j det(M_1j), where M_1j is the minor formed by removing row 1 and column j.

For a 2 x 2 minor, determinant equals ad - bc. The calculator recursively evaluates larger minors until every branch reaches a 2 x 2 or 1 x 1 matrix.

How to Use This Calculator

Select a square matrix size from 2 x 2 to 5 x 5.
Enter each matrix value into the input grid.
Click Calculate Determinant to compute the result.
Review the determinant, expansion summary, and minor matrices.
Use the CSV button for data export.
Use the PDF button to save a print-ready copy.

Frequently Asked Questions

1. What does this calculator compute?

It computes the determinant of a square matrix by expanding along the first row and evaluating each minor and cofactor contribution.

2. Which matrix sizes are supported?

The tool supports square matrices from 2 x 2 through 5 x 5. That range keeps calculations detailed while staying practical for manual review.

3. Why use cofactors instead of row reduction?

Cofactor expansion clearly shows how each element, sign, and minor affects the final determinant. It is especially useful for teaching, proofs, and checking hand calculations.

4. What if my matrix contains decimals?

Decimals are accepted. The calculator uses floating-point arithmetic and formats results cleanly, so both whole numbers and decimal matrices can be evaluated.

5. What does a zero determinant mean?

A zero determinant means the matrix is singular. Its rows or columns are linearly dependent, and the matrix does not have an inverse.

6. Does the calculator show intermediate work?

Yes. It lists term contributions from the first-row expansion and shows the minor matrices used to construct cofactors.

7. Can I save my results?

Yes. You can download a CSV summary for spreadsheet use and save a PDF version through the print-friendly export button.

8. Is this suitable for students and teachers?

Yes. The layout, examples, and expansion summary make it useful for homework checks, lesson demonstrations, revision, and matrix practice.