Matrix Minor, Matrix Cofactor, Adjugate Matrix, and Formula for Matrix Inversion – Linear Algebra Tutorial – Fusion of Engineering, Control, Coding, Machine Learning, and Science

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

What is a minor of a matrix and how to calculate minors of a matrix
What is a cofactor of a matrix and how to calculate cofactors of a matrix
What is an adjugate matrix and how to calculate it
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 $M_{i,j}$ minor is equal to the determinant of a submatrix obtained by deleting the $i$ th row and $j$ th column of the matrix.

Alternative (not rigorous) definition of a minor: Consider the matrix $A$ with its $(i,j)$ entry denoted by $a_{i,j}$ . Then the minor $M_{i,j}$ corresponding to the entry $a_{i,j}$ is calculated as a determinant of a submatrix obtained by erasing the row and the column to which the entry $a_{i,j}$ 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 $M_{12}$ of the 3 by 3 matrix $A$ :

(1) $\begin{align*}A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix}\end{align*}$

Since we want to compute the minor $M_{12}$ , corresponding to the entry $a_{12}$ , we need to erase the row and the column to which this entry belongs. That is, we need to erase the row $1$ and the column $2$ . By doing this, we obtain a submatrix $A_{12}$ defined by

(2) $\begin{align*}A_{12}=\begin{bmatrix}a_{21} & a_{23} \\ a_{31} & a_{33} \end{bmatrix}\end{align*}$

Finally, the minor $M_{12}$ is defined as the determinant of this matrix:

(3) $\begin{align*}M_{12}=\det (A_{12})=a_{21}a_{33} - a_{23}a_{31}\end{align*}$

Here is a complete list of all minors of this matrix

(4) $\begin{align*}M_{11}= a_{22}a_{33}-a_{23}a_{32} \\M_{12}= a_{21}a_{33}-a_{23}a_{31} \\M_{13}= a_{21}a_{32}-a_{22}a_{31} \\M_{21}= a_{22}a_{33}-a_{23}a_{32} \\M_{22}= a_{11}a_{33}-a_{13}a_{31} \\M_{23}= a_{11}a_{32}-a_{12}a_{31} \\M_{31}= a_{12}a_{23}-a_{13}a_{22}\\M_{32}= a_{11}a_{23}-a_{13}a_{21}\\M_{33}= a_{11}a_{22}-a_{12}a_{21}\end{align*}$

Cofactor of a Matrix

Next, we will define a cofactor of a matrix

DEFINITION OF A COFACTOR OF A MATRIX: The $(i,j)$ cofactor of a matrix, denoted by $C_{i,j}$ , is defined by

(5) $\begin{align*}C_{i,j}=(-1)^{i+j}\cdot M_{i,j}\end{align*}$

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 $(i,j)$ of the entry $a_{i,j}$ , we add these subscripts together to obtain $i+j$ , and then we compute the sign of the minor as $(-1)^{i+j}$ . The following matrix is filled with the signs of the corresponding minors:

(6) $\begin{align*}\text{Cofactor matrix}=\begin{bmatrix} +M_{11} & -M_{12} & +M_{13} \\ -M_{21} & +M_{22} & -M_{23} \\ +M_{31} & -M_{32} & +M_{33} \end{bmatrix}\end{align*}$

where $M_{i,j}$ is the $(i,j)$ minor of the matrix $A$ . The matrix cofactors of our 3 by 3 matrix $A$ are listed below

(7) $\begin{align*}C_{11} & = a_{22}a_{33}-a_{23}a_{32} \\C_{12}& = -(a_{21}a_{33}-a_{23}a_{31}) \\C_{13}& = a_{21}a_{32}-a_{22}a_{31} \\C_{21}& = -(a_{22}a_{33}-a_{23}a_{32}) \\C_{22}& = a_{11}a_{33}-a_{13}a_{31} \\C_{23}& = -(a_{11}a_{32}-a_{12}a_{31}) \\C_{31}& = a_{12}a_{23}-a_{13}a_{22}\\C_{32}& = -(a_{11}a_{23}-a_{13}a_{21})\\C_{33}& = a_{11}a_{22}-a_{12}a_{21}\end{align*}$

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 $A$ :

(8) $\begin{align*}A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix}\end{align*}$

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

(9) $\begin{align*}C=\begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33}\end{bmatrix}\end{align*}$

where $C_{i,j}$ is the $(i,j)$ cofactor of the matrix $A$ .

DEFINITION OF THE ADJUGATE MATRIX: The adjugate matrix of the matrix $A$ , denoted by $\text{adj}(A)$ is defined by

(10) $\begin{align*}\text{adj}(A)=C^{T}\end{align*}$

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

(11) $\begin{align*}\text{adj}(A) = \begin{bmatrix} C_{11} & C_{21} & C_{31} \\ C_{12} & C_{22} & C_{32} \\ C_{13} & C_{23} & C_{33} \end{bmatrix}\end{align*}$

Adjugate Matrix and Matrix Inversion

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

(12) $\begin{align*}A^{-1}=\frac{1}{\det (A)} \text{adj}(A)\end{align*}$

Minor of a Matrix

Cofactor of a Matrix

Adjugate Matrix

Adjugate Matrix and Matrix Inversion

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