How do you adjoint a matrix?

To adjoint a matrix, you first calculate its minor matrix, then its cofactor matrix, and finally take the transpose of the cofactor matrix. This process yields the adjoint matrix, also known as the adjugate matrix.

What is the Adjoint Matrix?

The adjoint matrix, often denoted as adj(A), is a specific matrix derived from a square matrix A. It is a fundamental concept in linear algebra, particularly for calculating the inverse of a matrix. For a square matrix A, its inverse A⁻¹ can be found using the formula:

A⁻¹ = (1 / det(A)) * adj(A)

where det(A) is the determinant of A. The adjoint matrix essentially helps generalize the concept of an inverse to square matrices of any size.

Steps to Adjoint a Matrix

The process of finding the adjoint matrix involves three main sequential steps:

1. Calculate the Minor Matrix

The first step is to construct the minor matrix M. Each element M_ij of the minor matrix is the minor of the corresponding element a_ij in the original matrix A.

What is a Minor? The minor M_ij of an element a_ij is the determinant of the submatrix formed by deleting the i-th row and j-th column of the original matrix A.
Process for each element:
1. For each element a_ij in the original matrix A, conceptually remove the row i and column j that a_ij belongs to.
2. Calculate the determinant of the remaining submatrix. This calculated determinant is the minor M_ij.
3. Place this minor value M_ij at the i-th row and j-th column of the new minor matrix M.

Example: For a 3x3 matrix A:

A = | a11 a12 a13 |
    | a21 a22 a23 |
    | a31 a32 a33 |

To find M11 (the minor for a11), you remove the first row and first column of A:
M11 = det | a22 a23 |
| a32 a33 |

2. Form the Cofactor Matrix

Once you have the minor matrix M, the next step is to create the cofactor matrix C. Each element C_ij of the cofactor matrix is the cofactor of the corresponding element a_ij in the original matrix A.

What is a Cofactor? A cofactor C_ij is derived from its minor M_ij by applying a sign based on its position: C_ij = (-1)^(i+j) * M_ij.
Sign Pattern: The (-1)^(i+j) term results in an alternating sign pattern across the matrix, starting with a positive sign for the element in the first row and first column:
```
| + - + |
| - + - |
| + - + |
```
This pattern extends for larger matrices.
Process:
1. Take each minor M_ij from the minor matrix M.
2. Multiply M_ij by (-1)^(i+j).
3. Place this result C_ij at the i-th row and j-th column of the new cofactor matrix C.

Example (continuing from minor example):
If M11 was (a22*a33 - a23*a32), then C11 = (+1) * M11.
If M12 was (a21*a33 - a23*a31), then C12 = (-1) * M12.

3. Transpose the Cofactor Matrix to Get the Adjoint

The final step is to find the transpose of the cofactor matrix C. This transposed matrix is the adjoint matrix adj(A).

What is a Transpose? The transpose of a matrix C, denoted C^T, is obtained by interchanging its rows and columns. The element originally at C_ij in matrix C becomes the element at (C^T)_ji in the transposed matrix.
Process:
1. Take the cofactor matrix C.
2. Swap its rows and columns. The element C_ij becomes the element in position (j,i) of the adjoint matrix.

Example (completing the process):
If the cofactor matrix C is:

C = | C11 C12 C13 |
    | C21 C22 C23 |
    | C31 C32 C33 |

Then the adjoint matrix adj(A) will be:

adj(A) = C^T = | C11 C21 C31 |
                 | C12 C22 C32 |
                 | C13 C23 C33 |

Practical Example: Finding the Adjoint of a 3x3 Matrix

Let's find the adjoint of the matrix A:

A = | 1 2 3 |
    | 0 4 1 |
    | 5 2 2 |

1. Calculate the Minor Matrix (M)

M11 = det | 4 1 | = (4*2 - 1*2) = 6
| 2 2 |
M12 = det | 0 1 | = (0*2 - 1*5) = -5
| 5 2 |
M13 = det | 0 4 | = (0*2 - 4*5) = -20
| 5 2 |
M21 = det | 2 3 | = (2*2 - 3*2) = -2
| 2 2 |
M22 = det | 1 3 | = (1*2 - 3*5) = -13
| 5 2 |
M23 = det | 1 2 | = (1*2 - 2*5) = -8
| 5 2 |
M31 = det | 2 3 | = (2*1 - 3*4) = -10
| 4 1 |
M32 = det | 1 3 | = (1*1 - 3*0) = 1
| 0 1 |
M33 = det | 1 2 | = (1*4 - 2*0) = 4
| 0 4 |

The minor matrix M is:

M = |  6  -5 -20 |
    | -2 -13  -8 |
    | -10   1   4 |

2. Calculate the Cofactor Matrix (C)

Apply the alternating sign pattern | + - + | to the minor matrix M.
| - + - |
| + - + |

C11 = +M11 = 6
C12 = -M12 = -(-5) = 5
C13 = +M13 = -20
C21 = -M21 = -(-2) = 2
C22 = +M22 = -13
C23 = -M23 = -(-8) = 8
C31 = +M31 = -10
C32 = -M32 = -(1) = -1
C33 = +M33 = 4

The cofactor matrix C is:

C = |  6   5 -20 |
    |  2 -13   8 |
    | -10 -1   4 |

3. Transpose the Cofactor Matrix to get the Adjoint (adj(A))

Swap the rows and columns of C:

adj(A) = C^T = |  6   2 -10 |
                |  5 -13  -1 |
                | -20  8   4 |

Applications of the Adjoint Matrix

Beyond finding the inverse, the adjoint matrix is also integral to:

Cramer's Rule: A method for solving systems of linear equations using determinants.
Characteristic Polynomial: The adjoint can be used in the computation of the characteristic polynomial of a matrix.
Theoretical Mathematics: It appears in various theorems and definitions within abstract algebra and linear transformations.

Summary of Adjoint Calculation Steps

Step	Description	Intermediate Result
1	Minor Matrix (M): For each element `a_ij`, find the determinant of the submatrix formed by removing row `i` and column `j`.	A matrix of all minors `M_ij`.
2	Cofactor Matrix (C): Apply the sign pattern `(-1)^(i+j)` to each minor `M_ij` to get `C_ij`.	A matrix of all cofactors `C_ij`.
3	Adjoint Matrix (adj(A)): Take the transpose of the cofactor matrix `C`.	The final `adj(A)` which is `C^T`.