Inverse of a Matrix and Its Applications, Objectives, Determinant of a Square Matrix, Minors and Cofactors of the Elements of Square Matrix, Adjoint of a Square Matrix

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Objectives

After studying this lesson, you will be able to:

  • Define a minor and a cofactor of an element of a matrix;

  • Find minor and cofactor of an element of a matrix;

  • Find the adjoint of a matrix;

  • Define and identify singular and non-singular matrices;

  • Find the inverse of a matrix, if it exists;

  • Represent system of linear equations in the matrix form ; and

  • Solve a system of linear equations by matrix method.

Determinant of a Square Matrix

We have already learnt that with each square matrix, a determinant is associated. For any given matrix, say its determinant will be . It is denoted by .

Similarly, for the matrix , the corresponding determinant is

Example:

Determine whether matrix A is singular or non-singular where

Solution:

Here,

Therefore, the given matrix is a singular matrix.

Minors and Cofactors of the Elements of Square Matrix

Consider a matrix

The determinant of the matrix obtained by deleting the row and column of , is called the minor of and is denotes by .

Cofactor of is defined as

For example, Minor of and Cofactor of

Example:

Find the minors and the cofactors of the elements of matrix

Solution:

For matrix ,

(minor of ) ;

(minor of ) ;

(minor of ) ;

(minor of ) ;

Adjoint of a Square Matrix

Let be a matrix. Then

Let and be the minor and cofactor of respectively. Then

We replace each element of by its cofactor and get

The transpose of the matrix of cofactors obtained in above is

The matrix obtained above is called the adjoint of matrix . It is denoted by .

Thus, adjoint of a given matrix is the transpose of the matrix whose elements are the cofactors of the elements of the given matrix.

Example:

Find the adjoint of

Solution:

Here, Let be the cofactor of the element .

Then,

We replace each element of by its cofactor to obtain its matrix of cofactors as

Transpose of matrix in is .

Thus,

Developed by: