Cofactor Expansion Calculator

Evaluate a 3×3 determinant by Laplace expansion and inspect every first-row minor, cofactor, and product.

3×3 determinant by cofactor expansion
Enter all nine matrix entries to expand along the first row.

About cofactor expansion

Cofactor expansion, also known as Laplace expansion, expresses a square matrix determinant as a weighted sum of smaller determinants. Select any row or column, multiply each entry by its signed minor, and add the products. This calculator uses the first row of a 3×3 matrix, making every stage easy to inspect while producing the same determinant that expansion along any other row or column would give. To find the minor of an entry, delete that entry's row and column. In a 3×3 matrix, the remaining values form a 2×2 matrix whose determinant is the upper-left times lower-right product minus the upper-right times lower-left product. A cofactor adds an alternating sign to the minor. Across the first row, the signs follow plus, minus, plus. The displayed cofactor values already include those signs. The determinant is then the first entry times its cofactor, plus the second entry times its cofactor, plus the third entry times its cofactor. Although the operation is described as addition, a negative cofactor naturally creates subtraction. Showing the minors, signed cofactors, and final products separately helps locate arithmetic mistakes and makes the result suitable for checking hand calculations. A zero determinant means the matrix is singular. Its rows or columns are linearly dependent, it has no multiplicative inverse, and the associated linear transformation collapses dimension. A nonzero determinant indicates an invertible matrix. The sign describes orientation, while the determinant's absolute value is the scale factor for area or volume under the corresponding linear transformation. Cofactor expansion generalizes to matrices of every square size, recursively reducing an n-by-n determinant to determinants of size n minus one. For large dense matrices, elimination methods are computationally faster, but Laplace expansion remains especially useful when a row or column contains many zeros. It is also fundamental to defining adjugates, inverses, and characteristic polynomials. This focused 3×3 calculator supports integers and decimals, preserves precision during calculation, and exposes the complete first-row expansion rather than returning an unexplained final number.

Cofactor expansion examples

These matrices demonstrate general, diagonal, and singular determinants.

Matrix rowsDeterminantExpansion insight
[1 2 3], [0 1 4], [5 6 0]1The signed first-row cofactors are -24, 20, and -5.
[2 0 0], [0 3 0], [0 0 4]24Only the first expansion product is nonzero.
[1 2 3], [2 4 6], [0 1 1]0The second row depends on the first, so the matrix is singular.

How to use cofactor expansion

  1. Enter the nine matrix values row by row.
  2. Select Calculate Cofactor Expansion.
  3. Review the three 2×2 minors and their alternating signed cofactors.
  4. Add the displayed expansion products to verify the determinant.

Cofactor expansion FAQ

What is the difference between a minor and a cofactor?

A minor is the determinant left after deleting one row and column. A cofactor is that minor multiplied by its checkerboard sign.

Can I expand along another row or column?

Yes, every complete row or column expansion produces the same determinant. Choosing one with several zeros usually reduces the arithmetic.

Why do cofactor signs alternate?

The sign at row i and column j is determined by whether i plus j is even or odd. This checkerboard pattern preserves determinant orientation and antisymmetry.

What does a zero determinant mean?

A zero determinant means the matrix is singular and has no inverse. Its rows or columns contain a linear dependency.

Does the calculator accept decimals?

Yes, every cell accepts integer or decimal values. Results are calculated with double-precision arithmetic and rounded only for display.