Laplace expansion by a selected column j:
det(A) = Σ aijCij, for i = 1 to n
Cij = (−1)i+j det(Mij)
Contribution of row i = aij × Cij
The calculator removes one row and the chosen column to build each minor matrix Mij. It then finds the minor determinant, applies the alternating sign, forms the cofactor, and adds every contribution to get the final determinant.
Choosing a column with zeros usually reduces work because zero entries add zero contribution.
- Choose a square matrix size from 2 × 2 up to 4 × 4.
- Pick the column you want to use for Laplace expansion.
- Enter matrix values, including decimals, negatives, or zeros.
- Set the decimal precision for displayed results.
- Click Calculate Determinant to show the answer under the header and above the form.
- Review the summary table, row-by-row cofactors, and minor matrices.
- Use the Plotly graph to compare each row contribution visually.
- Download the result data as CSV or PDF when needed.
| Example | Matrix | Chosen column | Determinant | Note |
|---|---|---|---|---|
| Case 1 | [ [3, 2], [5, 7] ] | 1 | 11 | Simple 2 × 2 validation case. |
| Case 2 | [ [2, 1, 3], [0, 4, 5], [1, -2, 6] ] | 2 | 61 | Good example for sign changes. |
| Case 3 | [ [1, 0, 2, 1], [3, 1, 0, 2], [4, -1, 2, 0], [2, 3, 1, 1] ] | 4 | 51 | Useful 4 × 4 practice matrix. |
1) What does expansion by column mean?
It means finding the determinant by selecting one column, building a cofactor for each entry in that column, and summing the resulting contributions.
2) Which column should I choose?
Any column works. A column containing several zeros is often best because zero entries produce zero contribution and shorten the calculation.
3) Does choosing another column change the determinant?
No. Every valid row or column expansion gives the same determinant. Only the amount of arithmetic work changes.
4) Why do cofactors use alternating signs?
The checkerboard sign pattern comes from the cofactor definition, (−1)i+j. It preserves the correct orientation and algebraic structure of determinant expansion.
5) Can this calculator handle decimals and negative values?
Yes. You can enter integers, decimals, negative numbers, and zeros. The calculator formats the output using your chosen decimal precision.
6) Why are some contributions equal to zero?
A contribution becomes zero when the selected entry is zero or when the corresponding cofactor equals zero. Both situations are common in sparse matrices.
7) What does a zero determinant mean?
A zero determinant means the matrix is singular. Its rows or columns are linearly dependent, so the matrix has no inverse.
8) How is the graph useful?
The graph shows which row terms dominate the selected column expansion. It helps you inspect positive, negative, and zero contributions quickly.