MINORS AND COFACTORS

Matrices and determinants are the most important topics for students as they have many applications. Understanding and knowing the concepts of minors and cofactors is crucial and necessary to compute the adjoint and the inverse of the matrix (or matrices). 

The minor of a matrix is defined as the value from the computation of a determinant of a square matrix, obtained by eliminating the row and the column which corresponds to the element under consideration.

The technique by which one can find cofactors of a matrix is known as cofactor expansion. If you wish to see the determinants of a large square matrix, you first need to find the minor of a matrix followed by its cofactors. 

INTRODUCTION TO THE CONCEPT OF MATRIX AND DETERMINANT OF A MATRIX.

• THE CONCEPT OF MATRIX.

• The concept of matrix has evolved a long way in an attempt to obtain methods that are compact as well as simple to solve systems of linear equations.
• A matrix can be defined as a well-ordered rectangular array of numbers (or sometimes variables) or functions.
• A matrix is denoted by a capital alphabet representing the rectangular array.
• Suppose m number of rows and n number of columns are present in a matrix then the number of entries present in the matrix will be m x n.
• Each entry in the matrix has two subscripts, one of them represents the position of the column and another represents the position of the row in the matrix like Aij where i means the position of the row while j means the position of the column.
• Some examples of a matrix are –

Row matrix, A = [1  17  90  32]1×4 

Column matrix, 

B = [ 3 ]

[ 4 ]

[-6 ]3×1

• A matrix is considered a square matrix if the number of rows and columns are the same.
• Therefore, aij m n matrix, where m = n, is a square matrix.
• For example, A = [ 2 3 4 ] 

           [ 5 6 7 ]

           [ 8 9 1 ]3×3

• THE CONCEPT OF DETERMINANT OF A MATRIX.

• A determinant can only be found if the given matrix is a square matrix.
• For any square matrix A = [aij] of order m, a real or complex number can be associated such that aij = (i, j)th element of A, known as the determinant of a matrix.
• Example: For a matrix,

A = [a b]

[c d]

Its determinant is expressed as,

|A| = |a b| = det (A).

    |c d|

APPLICATION OF MATRICES AND DETERMINANTS.

• Matrices and determinants are integral parts to solve complex problems. Because of this feature, it is widely used in the science field. 
• They help the scientist to put a lot of complex data together. They are vital in physics and engineering, where people have formulas based on multi-dimensional quantities. 
• Some fields where we can use matrices and determinants:

• Robotics
• Genetics 
• Statistics 
• Linear programming 
• Optimization 
• Intersection plans

THE CONCEPT OF MINOR AND COFACTORS

WHAT ARE MINORS?

• The minor of a matrix is defined as the value from the computation of a determinant of a square matrix, obtained by eliminating the row and the column which corresponds to the element under consideration. 
• The minor of a matrix of element aij is denoted by Mij.
• For example: 

The minor of 6 in the matrix

A = [1 2 3]

       [4 5 6]

       [7 8 9]

Is given by:

M23 = |1 2|

           |7 8|

= 8 – 14

= -6.

WHAT ARE COFACTORS?

• The cofactor of an element is the signed minor of that very element. 
• The cofactor of element aij is denoted by Aij and is expressed as –

Aij = (-1)i+j Mij

• For example:

Cofactors of all elements in the determinant

|A| = |1 -2|

           |4  3|

Is given by 

A11 = (-1)1+1 M11 = (-1)2 (3)  = 3

A12 = (-1)1+2 M12 = (-1)3 (4)  = -4

A21 = (-1)2+1 M21 = (-1)3 (-2) = 2

A22 = (-1)2+2 M22 = (-1)4 (1)  = 1

DIFFERENCE BETWEEN MINOR AND COFACTORS of Matrix?

The minor is the value in a matrix computed from the determinant obtained by eliminating the column and row the following matrix is in consideration. The determinant thus obtained is the minor. Now, for minors, we don’t consider the sign nomenclature based on the position of elements used to determine minors. At the same time, the cofactor of an element is the signed minor. But in the case of cofactors, we even consider the signs of cofactors emerging from the positions as well. 

Conclusion:

Through this precise and concise article on minors and cofactors, CBSE students have been made familiar with the concepts of matrix, determinant, applications of matrices and determinants, minor and cofactor of matrix, denoting minors and cofactors, and difference between minor and cofactor of a matrix.

Students now know that the minor of a matrix is defined as the value from the computation of a determinant of a square matrix, obtained by eliminating the row and the column which corresponds to the element under consideration, and the cofactor of an element is the signed minor of that very element.