Cofactors, Adjoints, Inverse of a Matrix

Cofactors

The cofactor of an element of a matrix is defined as , where is the minor of . The minor is the determinant of the matrix obtained by deleting the -th row and -th column from .

Example: Cofactors of a 2x2 Matrix

Consider the 2x2 matrix

The cofactors are calculated as:

So, the cofactor matrix is:

Example: Cofactors of a 3x3 Matrix

Consider the 3x3 matrix

The cofactors are:

The cofactor matrix is:

Adjoint

The adjoint of a matrix (or adjugate) is the transpose of the cofactor matrix. It is denoted as .

Example: Adjoint of a 2x2 Matrix

Using the cofactor matrix from the previous example:

The adjoint is:

Example: Adjoint of a 3x3 Matrix

Using the cofactor matrix:

The adjoint is:

Inverse

The inverse of a matrix , if it exists, is given by:

Example: Inverse of a 2x2 Matrix

For the matrix

the determinant is calculated as:

Using the adjoint from the previous example:

the inverse is:

Example: Inverse of a 3x3 Matrix

For the matrix

the determinant is calculated as:

Using the adjoint from the previous example:

the inverse is:

CONCEPTS_ML

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