Matrix Minor, Matrix Cofactor, Adjugate Matrix, and Formula for Matrix Inversion – Linear Algebra Tutorial

In this calculus and linear algebra tutorial, we will learn the following topics

1. What is a minor of a matrix and how to calculate minors of a matrix
2. What is a cofactor of a matrix and how to calculate cofactors of a matrix
3. What is an adjugate matrix and how to calculate it
4. How to invert a matrix by computing minors, cofactors, and matrix adjugate

In the follow-up tutorials, we explain how to use these definitions to compute the matrix inverse. In this tutorial, we will just state the general formula of matrix inverse in terms of minors, cofactors, and matrix adjugate.

Minor of a Matrix

DEFINITION OF A MINOR OF A MATRIX: For a square matrix, the minor is equal to the determinant of a submatrix obtained by deleting the th row and th column of the matrix.

Alternative (not rigorous) definition of a minor: Consider the matrix with its entry denoted by . Then the minor corresponding to the entry is calculated as a determinant of a submatrix obtained by erasing the row and the column to which the entry belong to.

Now, let us learn how to calculate a minor of a matrix. The calculation of the minor is illustrated in the figure below.

In the figure above, we illustrate how to compute the minor of the 3 by 3 matrix :

(1)  

Since we want to compute the minor , corresponding to the entry , we need to erase the row and the column to which this entry belongs. That is, we need to erase the row and the column . By doing this, we obtain a submatrix defined by

(2)  

Finally, the minor is defined as the determinant of this matrix:

(3)  

Here is a complete list of all minors of this matrix

(4)  

Cofactor of a Matrix

Next, we will define a cofactor of a matrix

DEFINITION OF A COFACTOR OF A MATRIX: The cofactor of a matrix, denoted by , is defined by

(5)  

From this definition, we see that the cofactor corresponding to the minor is simply computed as a signed minor. That is, we take the subscripts of the entry , we add these subscripts together to obtain , and then we compute the sign of the minor as . The following matrix is filled with the signs of the corresponding minors:

(6)  

where is the minor of the matrix . The matrix cofactors of our 3 by 3 matrix are listed below

(7)  

Adjugate Matrix

To define an adjugate matrix of the matrix A, we first need to define a matrix of cofactors. For clarity, we will state the definition for a 3 by 3 matrix. The definitions straightforwardly generalize for matrices of arbitrary dimensions. Consider again the matrix :

(8)  

For such a matrix, we can define the cofactor matrix as follows

(9)  

where is the cofactor of the matrix .

DEFINITION OF THE ADJUGATE MATRIX: The adjugate matrix of the matrix , denoted by is defined by

(10)  

that is, the adjugate matrix of the matrix is simply computed by transposing the matrix of cofactors. From (9), we have

(11)  

Adjugate Matrix and Matrix Inversion

The formula for matrix inversion by using the adjugate matrix is given by

(12)