In linear algebra, a minor of a matrix is the determinant of some smaller square matrix generated from by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are useful for calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition.

Definition and illustration

First minors

If is a square matrix, then the minor of the entry in the -th row and -th column (also called the minor, or a first minor) is the determinant of the submatrix formed by deleting the -th row and -th column. This number is often denoted . The cofactor is obtained by multiplying the minor by , and is often denoted .

To illustrate these definitions, consider the following matrix,

<math display=block>\begin{bmatrix}

1 & 4 & 7 \\

3 & 0 & 5 \\

-1 & 9 & 11 \\

\end{bmatrix}</math>

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

<math display=block> M_{2,3} = \det \begin{bmatrix}

1 & 4 & \Box \\

\Box & \Box & \Box \\

-1 & 9 & \Box \\

\end{bmatrix}= \det \begin{bmatrix}

1 & 4 \\

-1 & 9 \\

\end{bmatrix} = 9-(-4) = 13</math>

So the cofactor of the entry is

<math display=block>C_{2,3} = (-1)^{2+3}(M_{2,3}) = -13.</math>

General definition

Let be an matrix and an integer with , and . A minor of , also called minor determinant of order of or, if , the th minor determinant of (the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a matrix obtained from by deleting rows and columns. Sometimes the term is used to refer to the matrix obtained from as above (by deleting rows and columns), but this matrix should be referred to as a (square) submatrix of , leaving the term "minor" to refer to the determinant of this matrix. For a matrix as above, there are a total of <math display="inline">{m \choose k} \cdot {n \choose k}</math> minors of size . The minor of order zero is often defined to be 1. For a square matrix, the zeroth minor is just the determinant of the matrix. mean the determinant of the matrix that is formed as above, by taking the elements of the original matrix from the rows whose indexes are in and columns whose indexes are in , whereas some other authors mean by a minor associated to and the determinant of the matrix formed from the original matrix by deleting the rows in and columns in ;

Applications of minors and cofactors

Cofactor expansion of the determinant

The cofactors feature prominently in Laplace's formula for the expansion of determinants, which is a method of computing larger determinants in terms of smaller ones. Given an matrix , the determinant of , denoted , can be written as the sum of the cofactors of any row or column of the matrix multiplied by the entries that generated them. In other words, defining <math>C_{ij} = (-1)^{i+j} M_{ij}</math> then the cofactor expansion along the -th column gives:

<math display=block>\begin{align}

\det(\mathbf A) &= a_{1j}C_{1j} + a_{2j}C_{2j} + a_{3j}C_{3j} + \cdots + a_{nj}C_{nj} \\[2pt]

&= \sum_{i=1}^{n} a_{ij} C_{ij} \\[2pt]

&= \sum_{i=1}^{n} a_{ij}(-1)^{i+j} M_{ij}

\end{align}</math>

The cofactor expansion along the -th row gives:

<math display=block>\begin{align}

\det(\mathbf A) &= a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3} + \cdots + a_{in}C_{in} \\[2pt]

&= \sum_{j=1}^{n} a_{ij} C_{ij} \\[2pt]

&= \sum_{j=1}^{n} a_{ij} (-1)^{i+j} M_{ij}

\end{align}</math>

Inverse of a matrix

One can write down the inverse of an invertible matrix by computing its cofactors by using Cramer's rule, as follows. The matrix formed by all of the cofactors of a square matrix is called the cofactor matrix (also called the matrix of cofactors or, sometimes, comatrix):

<math display=block>\mathbf C = \begin{bmatrix}

C_{11} & C_{12} & \cdots & C_{1n} \\

C_{21} & C_{22} & \cdots & C_{2n} \\

\vdots & \vdots & \ddots & \vdots \\

C_{n1} & C_{n2} & \cdots & C_{nn}

\end{bmatrix} </math>

Then the inverse of is the transpose of the cofactor matrix times the reciprocal of the determinant of :

<math display=block>\mathbf A^{-1} = \frac{1}{\operatorname{det}(\mathbf A)} \mathbf C^\mathsf{T}.</math>

The transpose of the cofactor matrix is called the adjugate matrix (also called the classical adjoint) of .

The above formula can be generalized as follows: Let

<math display=block>\begin{align}

I &= 1 \le i_1 < i_2 < \ldots < i_k \le n, \\[2pt]

J &= 1 \le j_1 < j_2 < \ldots < j_k \le n,

\end{align}</math>

be ordered sequences (in natural order) of indexes (here is an matrix). Then

<math display=block>[\mathbf A^{-1}]_{I,J} = \pm\frac{[\mathbf A]_{J',I'{\det \mathbf A},</math>

where denote the ordered sequences of indices (the indices are in natural order of magnitude, as above) complementary to , so that every index appears exactly once in either or , but not in both (similarly for the and ) and denotes the determinant of the submatrix of formed by choosing the rows of the index set and columns of index set . Also, <math>[\mathbf A]_{I,J} = \det \bigl( (A_{i_p, j_q})_{p,q = 1, \ldots, k} \bigr).</math> A simple proof can be given using the wedge product. Indeed,

<math display=block>\bigl[ \mathbf A^{-1} \bigr]_{I,J} (e_1\wedge\ldots \wedge e_n) = \pm(\mathbf A^{-1}e_{j_1})\wedge \ldots \wedge(\mathbf A^{-1}e_{j_k})\wedge e_{i'_1}\wedge\ldots \wedge e_{i'_{n-k, </math>

where <math>e_1, \ldots, e_n</math> are the basis vectors. Acting by on both sides, one gets

<math display=block>\begin{align}

&\ \bigl[\mathbf A^{-1} \bigr]_{I,J} \det \mathbf A (e_1\wedge\ldots \wedge e_n) \\[2pt]

=&\ \pm (e_{j_1})\wedge \ldots \wedge(e_{j_k})\wedge (\mathbf A e_{i'_1})\wedge\ldots \wedge (\mathbf A e_{i'_{n-k) \\[2pt]

=&\ \pm [\mathbf A]_{J',I'}(e_1\wedge\ldots \wedge e_n).

\end{align}</math>

The sign can be worked out to be

<math display=block>(-1)^{\left( \sum_{s=1}^{k} i_s - \sum_{s=1}^{k} j_s \right)},</math>

so the sign is determined by the sums of elements in and .

Other applications

Given an matrix with real entries (or entries from any other field) and rank , then there exists at least one non-zero minor, while all larger minors are zero.

We will use the following notation for minors: if is an matrix, is a subset of with elements, and is a subset of with elements, then we write for the minor of that corresponds to the rows with index in and the columns with index in .

  • If is square and , then is called a principal minor.
  • If is square and , then the principal minor is called a leading principal minor (of order ) or corner (principal) minor (of order ). For an square matrix, there are leading principal minors.
  • A basic minor of a matrix with rank is an minor with nonzero value. Moreover, it is denoted as and defined in the same way as cofactor:

<math display=block>\mathbf{A}_{ij} = (-1)^{i+j} \mathbf{M}_{ij}</math>

Using this notation the inverse matrix is written this way:

<math display=block>\mathbf{M}^{-1} = \frac{1}{\det(M)}\begin{bmatrix}

A_{11} & A_{21} & \cdots & A_{n1} \\

A_{12} & A_{22} & \cdots & A_{n2} \\

\vdots & \vdots & \ddots & \vdots \\

A_{1n} & A_{2n} & \cdots & A_{nn}

\end{bmatrix} </math>

Keep in mind that adjunct is not adjugate or adjoint. In modern terminology, the "adjoint" of a matrix most often refers to the corresponding adjoint operator.

See also

  • Submatrix
  • Compound matrix

References

  • MIT Linear Algebra Lecture on Cofactors at Google Video, from MIT OpenCourseWare