Minors and Cofactors

When working with matrices larger than 2×2, computing determinants requires a systematic method that breaks the problem into smaller, more manageable pieces. Minors and cofactors provide exactly this framework. The minor of an entry in a matrix is the determinant of the smaller matrix formed by deleting the row and column containing that entry. The cofactor is simply the minor with a sign attached — either positive or negative depending on the position of the entry in the matrix. Together, minors and cofactors form the backbone of cofactor expansion (also called Laplace expansion), a recursive algorithm that reduces the determinant of any n×n matrix to a combination of (n−1)×(n−1) determinants. For a 3×3 matrix this means computing three 2×2 determinants; for a 4×4 matrix it means four 3×3 determinants, and so on. This recursive structure makes determinants computable for matrices of any size. Beyond computing determinants, cofactors appear in one of the most elegant formulas in linear algebra: the adjugate (or classical adjoint) of a matrix, which leads directly to a formula for the matrix inverse. Understanding minors and cofactors therefore unlocks not only determinant computation but also deeper insights into how matrices encode geometric transformations, areas, and volumes.

Key takeaways
  • Defining the Minor M_{ij}
  • Attaching the Sign: The Cofactor C_{ij}
  • Cofactor Expansion Along a Row or Column
  • The Adjugate Matrix and the Inverse Formula
  • Why the Formula Works: The Key Identity

Defining the Minor M_{ij}

Given an n×n matrix A, the (i,j) minor, written M_{ij}, is the determinant of the (n−1)×(n−1) matrix obtained by deleting row i and column j from A. For example, in a 3×3 matrix A, the minor M_{12} is the determinant of the 2×2 matrix left after removing row 1 and column 2. The minor is always a scalar (a number), not a matrix.

Attaching the Sign: The Cofactor C_{ij}

The (i,j) cofactor is defined as C_{ij} = (−1)^{i+j} M_{ij}. The factor (−1)^{i+j} creates a checkerboard pattern of plus and minus signs across the matrix: positions where i+j is even get a plus sign, and positions where i+j is odd get a minus sign. For a 3×3 matrix the sign pattern is: + − + / − + − / + − +. This sign convention is essential — using the wrong sign gives the wrong determinant.

Cofactor Expansion Along a Row or Column

The determinant of A can be computed by expanding along any row i: det(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + ⋯ + a_{in}C_{in}. Equivalently, expanding along any column j gives: det(A) = a_{1j}C_{1j} + a_{2j}C_{2j} + ⋯ + a_{nj}C_{nj}. Both expansions give the same result. Strategically choosing a row or column with many zeros minimizes computation — each zero entry contributes nothing to the sum, so its cofactor need not be computed.

The Adjugate Matrix and the Inverse Formula

The cofactor matrix of A is the n×n matrix whose (i,j) entry is C_{ij}. The adjugate (or adjoint) of A, written adj(A), is the transpose of the cofactor matrix: adj(A) = [C_{ij}]^T, meaning the (i,j) entry of adj(A) is C_{ji}. The powerful inverse formula states: A^{−1} = (1/det(A)) · adj(A), valid whenever det(A) ≠ 0. This gives an explicit, closed-form expression for the inverse of any invertible matrix.

Why the Formula Works: The Key Identity

The reason the adjugate formula works is the identity A · adj(A) = det(A) · I, where I is the identity matrix. To see why: the (i,i) diagonal entries of the product equal a_{i1}C_{i1} + ⋯ + a_{in}C_{in} = det(A) by cofactor expansion. The off-diagonal entries equal zero because they represent cofactor expansions of a matrix with two identical rows, which always has determinant zero. Dividing both sides by det(A) gives A · A^{−1} = I.

The Key Equation

det⁡(A)=∑j=1naij Cij=∑j=1naij (−1)i+j Mij\det(A) = \sum_{j=1}^{n} a_{ij}\,C_{ij} = \sum_{j=1}^{n} a_{ij}\,(-1)^{i+j}\,M_{ij}
  • a_{ij} — Matrix entry (dimensionless): The element of matrix A located in row i and column j.
  • C_{ij} — Cofactor (dimensionless): The signed minor: C_{ij} = (−1)^{i+j} M_{ij}. Positive when i+j is even, negative when i+j is odd.
  • M_{ij} — Minor (dimensionless): The determinant of the submatrix obtained by deleting row i and column j from A.
  • (-1)^{i+j} — Sign factor (dimensionless): Creates the checkerboard pattern of signs. Equals +1 if i+j is even, −1 if i+j is odd.

Worked Example

Problem. Compute the determinant of A = [[3, 1, 2], [0, −1, 4], [5, 2, −3]] using cofactor expansion along the first column.

det⁡(A)=3(−5)+0+5(6)=−15+30=15\det(A) = 3(-5) + 0 + 5(6) = -15 + 30 = 15

Answer. det(A) = 15

Figure
Diagram
Sign Pattern (−1)^(i+j) for a 3×3 Matrix+i+j=2−i+j=3+i+j=4−i+j=3+i+j=4−i+j=5+i+j=4−i+j=5+i+j=6i=1i=2i=3j=1j=2j=3Green = positive cofactor sign (+1), Red = negative cofactor sign (−1)
Origins · 1 minPierre-Simon Laplace and how this was discovered
PL
Pierre-Simon Laplace
1749–1827 · French

Laplace gave a rigorous formulation of cofactor expansion for computing determinants, which is why the method is also called Laplace expansion. While Leibniz and Cramer had earlier worked with determinants, Laplace systematized and proved the general theorem that the determinant of any n×n matrix can be expanded along any row or column using cofactors, publishing this result in 1772. Laplace is equally famous in probability and astronomy. Napoleon Bonaparte reportedly asked him where God fit into his celestial mechanics, to which Laplace replied: 'I had no need of that hypothesis.'

  • Determinant of a 2×2 Matrix — The 2×2 determinant ad − bc is the base case for cofactor expansion — every minor in a 3×3 expansion reduces to a 2×2 determinant.
  • Properties of Determinants — Properties such as row operations changing the determinant underpin why cofactor expansion along a zero-heavy row is computationally efficient.
  • The Inverse of a Matrix — The adjugate formula A⁻¹ = adj(A)/det(A) directly connects cofactors to finding matrix inverses.
  • Cramer's Rule — Cramer's Rule uses determinants and cofactors to solve systems of linear equations explicitly, building directly on cofactor expansion.