Minor, Cofactor, Adjoint Matrix

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To understand the cofactor matrix, we first need to introduce the concept of a minor.

For a square matrix $A$

The minor $M_{ij}$ of an element $A_{ij}$ is the determinant of the submatrix obtained by deleting the $i$th row, $j$th column from $A$.

Simple Example : $2{\times}2$ Matrix

Let's find the minor matrix of a $2{\times}2$ matrix

$$ A = \begin{bmatrix} 3 & 1\\ 2 & 4 \end{bmatrix} $$
For $M_{11}$, we remove row $n$ and column $n$ and find the determinant:
- $M_{11} = |[4]| = 4$
- $M_{12} = |[2]| = 2$
- $M_{21} = |[1]| = 1$
- $M_{22} = |[3]| = 3$
Therefore, the minor matrix without ($i$,$j$) is

$$ M = \begin{bmatrix} 4 & 2\\ 1 & 3 \end{bmatrix} $$

Advanced Example : $3{\times}3$ Matrix

Let's find the minor matrix of a $3{\times}3$ matrix

$$ A = \begin{bmatrix} 2 & 1 & 3\\ 0 & 4 & -1\\ 5 & 2 & 2 \end{bmatrix} $$
For example, to find $M_{11}$, we remove the first row and first column:

$$ M_{11} = \det\begin{bmatrix} 4 & -1\\ 2 & 2 \end{bmatrix} = (4 \times 2) - (-1 \times 2) = 10 $$
Following this process for all elements, we get the minor matrix.

$$ M = \begin{bmatrix} 10 & -10 & 8\\ 7 & -1 & -9\\ 5 & 5 & 7 \end{bmatrix} $$

What’s Cofactor?

The cofactor $C_{ij}$ is calculated by multiplying the minor $M_{ij}$ by a sign factor.

$$ C_{ij} = (-1)^{i+j} \cdot \det[M_{ij}] $$

The sign alternates depending on the position $(i,j)$ in the matrix.

$$ \begin{bmatrix}

Cofactor matrix is formed by replacing each element $A_{ij}$ of a square matrix $A$ with its corresponding cofactor $C_{ij}$.

For an $n{×}n$ matrix $A$, the cofactor matrix is denoted as it.

$$ C_A = [C_{ij}] $$

Example

Here is $4{\times}4$ matrix $A$

$$ A = \begin{bmatrix} 2 & -1 & 3 & 0\\ 1 & 4 & -2 & 5\\ 0 & 2 & 1 & 3\\ 3 & -3 & 2 & 1 \end{bmatrix} $$
For $C_{11}$, we first find the minor by calculating the determinant of the $3{\times}3$ matrix after removing.

$$ M_{11} = \det\begin{bmatrix} 4 & -2 & 5\\ 2 & 1 & 3\\ -3 & 2 & 1 \end{bmatrix} = 37 $$
Continuing this process for all elements, we get the minor matrix.

$$ M = \begin{bmatrix} 37 & -38 & 17 & 31\\ -28 & 17 & 13 & -7\\ -45 & 18 & 24 & -33\\ 5 & -31 & 7 & 23 \end{bmatrix} $$
Notice how the signs alternate according to the position ($i,j$).

$$ \begin{bmatrix}
- & - & + & -\\
- & + & - & +\\
- & - & + & -\\
- & + & - & + \end{bmatrix} $$
Following this process for applying the appropriate signs (+, -)

$$ C_A = \begin{bmatrix} 37 & 38 & 17 & -31\\ 28 & 17 & -13 & -7\\ -45 & -18 & 24 & 33\\ -5 & -31 & -7 & 23 \end{bmatrix} $$

Adjoint matrix is defined as the transpose of the cofactor matrix.

$$ \text{adj}(A) = C_A^T $$

This means that in the adjoint matrix, element $(i,j)$ corresponds to element $(j,i)$ from the cofactor matrix.

Example

Here is $4{\times}4$ matrix $A$.

$$ A = \begin{bmatrix} 2 & -1 & 3 & 0\\ 1 & 4 & -2 & 5\\ 0 & 2 & 1 & 3\\ 3 & -3 & 2 & 1 \end{bmatrix} $$
Here is the cofactor matrix of matrix $A$.

$$ C_A = \begin{bmatrix} 37 & 38 & 17 & -31\\ 28 & 17 & -13 & -7\\ -45 & -18 & 24 & 33\\ -5 & -31 & -7 & 23 \end{bmatrix} $$

$$ \text{adj}(A) = C_A^T = \begin{bmatrix} 37 & 28 & -45 & -5\\ 38 & 17 & -18 & -31\\ 17 & -13 & 24 & -7\\ -31 & -7 & 33 & 23 \end{bmatrix} $$

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Python makes it easy to calculate matrix minors, cofactors, adjoints. Let's implement each concept using the NumPy.

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