Cofactor Expansion Old Calculator - Laplace Expansion Solver

A cofactor expansion calculator turns a square matrix into a step-by-step Laplace expansion, so the determinant of a 2x2, 3x3, or 4x4 matrix becomes a sum of products a_ij * C_ij that you can read off one at a time. Use this cofactor expansion calculator when the matrix has zeros in a row or column, when you want to see the sign factors and the recursive sub-determinants, or when a homework problem asks you to show your work for a determinant by minors.

• • Determinant by minors: Compute the determinant of a 2x2, 3x3, or 4x4 matrix with a fully expanded Laplace sum, including the sign factors and the recursive sub-determinants.
• • Linear algebra homework: Show every term a_ij * C_ij in a single line so a homework solution, exam answer, or teaching slide displays the work alongside the final number.
• • Building a cofactor matrix: Read off individual cofactors C_ij, which are the building blocks of the classical adjoint and the inverse matrix formula A^(-1) = (1 / det A) * adj A.

Cofactor expansion is the most common way to introduce determinants in a linear-algebra course, and it remains a useful tool for small matrices. The same idea extends to the matrix inverse through the adjugate.

For a 2x2 matrix the expansion is two terms and one 1x1 minor. For a 3x3 matrix the expansion is three terms with 2x2 minors. For a 4x4 matrix each minor is itself a 3x3 cofactor expansion.

For a single matrix operation outside the determinant workflow, the matrix calculator is the closest general-purpose tool across the site.

The calculator reads the matrix entries, the chosen direction, and the chosen index, then for each entry on the selected line it builds the (n-1)x(n-1) submatrix, takes its determinant, multiplies by the sign factor (-1)^(i+j), and adds all the products together. That sum is det(A), and the same answer is reached regardless of which row or column you pick.

• Matrix A: The square matrix whose determinant we want. n can be 2, 3, or 4.
• Submatrix A_ij: The (n-1)x(n-1) matrix after removing row i and column j. Its determinant is the minor M_ij.
• Cofactor C_ij: C_ij = (-1)^(i+j) * M_ij. The sign factor alternates like a checkerboard.
• Term a_ij · C_ij: One product in the final sum. det(A) is built by adding these terms along the chosen line.

The 2x2 case is the base of the recursion: det [[a, b], [c, d]] = ad − bc, which is the cofactor expansion a·d + b·(−c). For 3x3 and 4x4 matrices the expansion uses two or three nested 2x2 minors.

The sign factor (-1)^(i+j) produces a checkerboard pattern across any row or column. Forgetting this factor is the most common mistake when computing cofactors by hand, so the cofactor expansion calculator surfaces the signed term in the sum formula.

According to Wikipedia, the cofactor of the (i,j)-th entry of an n×n matrix is defined as C_ij = (-1)^(i+j) times the determinant of the (n-1)×(n-1) submatrix obtained by deleting row i and column j.

Once the cofactors C_ij are in hand, the adjoint matrix calculator builds the classical adjoint by transposing the cofactor matrix, which is the next step toward the inverse matrix formula.

Four ideas appear in every cofactor expansion, so it helps to separate them before reading the result panel or trying a 4x4 example by hand.

For 3x3 and 4x4 matrices the same idea repeats, and the calculator hides the recursion by reporting each term a_ij * C_ij as a separate number.

For 3x3 inputs the cofactor expansion of det(A) is closely related to the cross product, and the cross product calculator handles the vector side of that relationship.

The calculator accepts integer, decimal, and negative entries. Unused entries are ignored.

1. 1 Pick the matrix size: Choose 2x2, 3x3, or 4x4. The matrix grid stays the same; the calculator just ignores entries outside the chosen size.
2. 2 Enter the matrix entries: Type the entries of A in row order. The labels a_11, a_12, ... match the standard matrix notation.
3. 3 Pick the expansion line: Set the direction (row or column) and the index (1..n). Pick the line with the most zeros to make the sum short.
4. 4 Read the determinant: The primary output is det(A), rounded to 6 decimal places. The same value comes out for any choice of row or column.
5. 5 Inspect each term a_ij · C_ij: The four term rows show the contribution of each entry on the chosen line. A zero term tells you that entry contributes nothing.

If A = [[2, 0, 1], [0, 3, 0], [1, 0, 4]] and you pick column 2, the calculator returns det(A) = 0 + 3·7 + 0 = 21 with the only non-zero term coming from a_22 = 3 and its cofactor C_22 = 7.

After the determinant is in hand, the characteristic polynomial calculator gives the same scalar through the trace-and-determinant shortcut for 3x3 matrices.

The main benefit of this cofactor expansion calculator is replacing a recursive determinant calculation with a single screen that shows the per-term products and the running total.

• • Shows the full Laplace sum: Every term a_ij * C_ij is visible, so a homework solution, a teaching slide, or a textbook exercise can display the work without retyping it.
• • Catches sign and indexing mistakes: The sign factor (-1)^(i+j) is easy to forget when the index is not (1,1). The calculator applies the sign automatically, and the sum formula shows the sign in the rendered string.
• • Lets you compare expansion strategies: Switch between rows and columns to confirm the answer is the same, and pick the line with the most zeros to keep the sum short.
• • Pairs with the adjugate and the inverse: The individual cofactors C_ij are the entries of the classical adjoint (after transposition), so the calculator is a clean starting point for an adjugate or inverse matrix calculation.

The 2x2 and 3x3 modes are quick checks for any small determinant in geometry or physics, and the 4x4 mode covers the largest matrix you usually want to expand by hand. For matrices larger than 4x4, row reduction is faster.

If the same matrix comes up again, the result panel is a one-line summary that you can copy and paste. Changing a single entry updates the sum formula and the final determinant.

Three matrix properties and two input choices drive the result, and the output changes in ways worth understanding before relying on the answer.

• • The calculator only handles 2x2, 3x3, and 4x4 matrices. For 5x5 and larger, Gaussian elimination is usually faster than cofactor expansion, and the number of terms grows as n!.
• • The output is the determinant computed by cofactor expansion, not the eigenvalues or the matrix inverse. To go from det(A) to the inverse, you still need the adjugate and the formula A^(-1) = (1 / det A) * adj A.
• • The result assumes real-valued matrix entries. If the entries are complex, the same formulas apply but the displayed sub-determinants should be read as complex values.

If the matrix has repeated rows or columns, the determinant is 0 and every line of the sum returns 0. The cofactor expansion is still correct, but the answer is uninformative because the matrix is not invertible.

According to Wolfram MathWorld, the determinant of a square matrix can be expanded along any row i or column j as a sum a_ij * C_ij, which is known as Laplace expansion.

According to Wikipedia Laplace expansion, Laplace expansion is recursive because computing the determinant of an n×n matrix reduces to computing the determinants of (n−1)×(n−1) submatrices, and a row with the most zeros is usually the most efficient choice.

When the matrix is symmetric and positive-definite, the Cholesky decomposition calculator avoids cofactor expansion entirely by writing A as L · L^T for a lower-triangular L.