20-06-2024 23:05

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Tags : mathematics Matrix linear algebra

Adjoint

let $A=[a_{ij}]$ be a matrix write all cofactors of the set $A$ and then write all of them in a matrix the transpose of the matrix $A_{ij}$ is called adjoint matrix

Let $A=\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ & & & \\ a_{21}& a_{22}& \cdots & a_{2n}\\ & & & \\ ⋮& ⋮& \ddots & ⋮\\ & & & \\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right)$ be an m x n square matrix.

The minor of element $a_{ij}$ is denoted by $M_{ij}$ and is the determinant of the submatrix obtained by deleting the $i^{th}$ row and $j^{th}$ column of A. We can express this using the following code:

M_{ij} = \det \begin{pmatrix} ... & a_{1j} & ... \\ \\ ... & a_{{i-1},j} & ... \\ \\ ... & a_{{i+1},j} & ... \\ \\ ... & a_{mj} & ... \end{pmatrix} \end{equation}$$ The cofactor of element $a_{ij}$ is denoted by $C_{ij}$ and is calculated as follows: $$\begin{equation} C_{ij} = (-1)^{i+j} \cdot M_{ij} \end{equation}

where $(-1{)}^{i+j}$ is 1 if the sum of i and j is even, and -1 if the sum is odd.

To find all the cofactors of matrix A, we can calculate the minor for each element and multiply it by the corresponding $(-1{)}^{i+j}$ term. These cofactors can then be arranged into a new matrix, called the cofactor matrix, denoted by $C_A$.

References