MATRICES AND MATRIX OPERATIONS: Unit 12

Dr. Wlodzislaw Kostecki

The Papua New Guinea University of Technology (PNGUT)

Department of Electrical and Communication Engineering

Lae, Morobe Province

Papua New Guinea

Copyright © 2000 by Wlodzislaw Kostecki

All rights reserved

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(12) The minor and cofactor of a matrix element

OBJECTIVES :

• To define the minor and cofactor of a matrix element.

• To provide alternative methods of computing the minor and cofactor.

• To introduce the Laplace expansion theorem and show its application.

• To introduce further functions from Maple ’ s main library that are useful in matrix computations.

> restart : with(linalg, coldim, delcols, delrows, det, minor, rowdim) :

Both the minor and cofactor of a matrix element are defined only for square matrices.

Consider a ( × ) matrix [ A ] given as

> A := matrix(3, 3, [a[11], a[12], a[13], a[21], a[22], a[23], a[31], a[32], a[33]]) : A = matrix(A) ;

A. The minor of a matrix element

The minor of an element in an ( × ) matrix is defined as the matrix of order ( ) × ( ) obtained by the deletion of row and column .

For example, choose the element in the row and column of the matrix [ A ] and obtain the minor of this element. This operation may be performed using either of the following alternative methods.

Method 1 . Using the delrows and delcols functions:

> `minor(a12)` := delcols(delrows(A, 1..1), 2..2) : Minor(a[12]) = matrix(`minor(a12)`) ;

Method 2 . Using the minor function:

> i := 1 : j := 2 : `minor(a12)` := minor(A, i, j) : Minor(a[12]) = matrix(`minor(a12)`) ;

The determinant of the minor of the element is

> `det(minor(a12))` := det(`minor(a12)`) : Det(Minor(a[12])) = `det(minor(a12))` ;

B. The cofactor of a matrix element

The cofactor or signed minor of an element in an ( × ) matrix is defined as the product of the determinant of the minor of this element and scalar .

For example, the cofactor of the above element is

> cofactor(a12) := (-1)^(i+j) * `det(minor(a12))` : Cofactor(a[12]) = cofactor(a12) ;

Using the cofactors of row or column elements of a matrix enables evaluation of the determinant of the matrix, according to the Laplace expansion theorem, which states:

"The determinant associated with any ( × ) matrix [ A ] is obtained by summing the products of the elements and their cofactors in any row or column of the matrix . "

If matrix [ A ] has the general element and the corresponding cofactor is , then this theorem may be expressed as follows:

• Expansion by elements of a row:

i = 1, 2, ..., n

• Expansion by elements of a column:

j = 1, 2, ..., n

* * *

N.B. Since the expansion of the determinant of a matrix can be done about any one row or any one column, it is convenient to choose a row or a column with as many as possible.

* * *

Choosing the first row ( ) of the above matrix [ A ] results in the following expansion of the determinant of [ A ] displayed in "like-in-a-book" form:

> i:=1 : for j to coldim(A) do `det(minor(A,i,j))`[i, j] := Det(minor(A, i, j)) : a[i, j] := a[cat(i, j)] : od :

> j := 'j' : `det(A)` := sum((-1)^(i+j)*a[i, j]*`det(minor(A,i,j))`[i, j], j=1..coldim(A)) : Det(A)=`det(A)` ;

The cat function is used above to c onc at enate together the indices of the elements into a string expression. The sum function performs the sum mation of the expressions under this function .

Evaluation of the determinant of matrix [ A ] yields the following expression:

> for j to coldim(A) do `det(minor(A,i,j))`[i, j] := det(minor(A, i, j)) : od :

> j := 'j' : `det(A)` := sum((-1)^(i+j)*a[i, j]*`det(minor(A,i,j))`[i, j], j=1..coldim(A)) : Det(A) = `det(A)` ;

Application of the normal , simplify , or expand function to the above result expands it to the following expression:

> `det(A)` := normal(`det(A)`) : Det(A) = `det(A)` ; a[1,2] := 'a[1,2]' :

* * *

N.B. Evaluation of the determinant of the matrix [ A ] can be done in Maple directly by applying the det function to the matrix [ A ], i.e.

> `det(A)` := sort(det(A)) : Det(A) = `det(A)` ;

The sorting function sort is used above to obtain a strict correspondence of the product terms in both results.

* * *

N.B. It follows from the Laplace expansion theorem that the sum of the products of the elements of any row (or column) of a square matrix with the cofactors corresponding to the elements of a different row (or column) is .

Exemplarily, consider a ( × ) matrix [ A ] given as

> A := matrix(3, 3, [1, 3, 2, 4, 1, 2, 3, 1, 3]) : A = matrix(A) ;

and verify that the sum of the products of the elements of column 1 with the corresponding cofactors of the elements of column 2 is .

(a) The cofactors of the elements of column are

> j := 2 : for i to rowdim(A) do cofactor(a[i, j]) := (-1)^(i+j)*det(minor(A, i, j)) : print(Cofactor(a[i, j]) = cofactor(a[i, j])) : od : i := 'i' :

(b) The sum of the products of the elements of column with the corresponding cofactors of the elements of column is

> `sum(A[i,1] cofactor(a[i,j])` := sum(A[i, 1] * cofactor(a[i, j]), i=1..rowdim(A)) :

> Sum(a[i, 1] * Cofactor(a[i, j]), i=1..rowdim(A)) = `sum(A[i,1] cofactor(a[i,j])` ;

* * *

Proceed to Unit (13) for " The adjoint of a matrix ".

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