Minor (linear algebra)

In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which in turn are useful for computing both the determinant and inverse of square matrices.

If A is a square matrix, then the minor of the entry in the i-th row and j-th column (also called the (i,j) minor, or a first minor) is the determinant of the submatrix formed by deleting the i-th row and j-th column. This number is often denoted M_i,j. The (i,j) cofactor is obtained by multiplying the minor by ${\displaystyle (-1)^{i+j}}$.

To illustrate these definitions, consider the following 3 by 3 matrix,

To compute the minor M₂₃ and the cofactor C₂₃, we find the determinant of the above matrix with row 2 and column 3 removed.

So the cofactor of the (2,3) entry is

Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n. A k × k minor of A, also called minor determinant of order k of A or, if ${\displaystyle m=n}$, (n-k):th minor determinant of A, with the word "determinant" often omitted and the word "order" sometimes replaced by "degree", is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns. Sometimes the term is used to refer to the k × k matrix obtained from A as above (by deleting m − k rows and n − k columns), but this matrix should be referred to as a (square) submatrix of A, leaving the term "minor" to refer to the determinant of this matrix. For a matrix A as above, there are a total of ${\displaystyle {m \choose k}\cdot {n \choose k}}$ minors of size k × k. Minor of order zero is often defined to be 1. For a square matrix, zeroth minor is just the determinant of the matrix.

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