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Adjugate matrix reminds me of the dark days before Git.
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When calculating matrix inverses, there are two main methods: the adjugate matrix method and the augmented matrix method. In this article, we’ll explore these approaches in detail and compare their advantages.
Finding Matrix Inverse Using Adjugate Matrix
Cofactor Formula
For an n×n matrix A = [aᵢⱼ], let’s define Mᵢⱼ as the minor matrix of A, obtained by deleting the i-th row and j-th column:
Mᵢⱼ = det(Aᵢⱼ)
The cofactor Cᵢⱼ is defined as:
Cᵢⱼ = (-1)ⁱ⁺ʲMᵢⱼ
Where(-1)ⁱ⁺ʲ accounts for the sign factor based on the element’s position.
Developer analogy: The cofactor is like calculating how much impact a change in one module has on other dependent modules.
Adjugate Matrix
The adjugate matrix of A (denoted as adj(A)) is defined as the matrix of cofactors:
adj(A) = [Cᵢⱼ]ₙₓₙ
For example:
⎛C₁₁ C₂₁ ⋯ Cₙ₁⎞
adj(A) = ⎜C₁₂ C₂₂ ⋯ Cₙ₂⎟
⎜ ⋮ ⋮ ⋱ ⋮ ⎟
⎝C₁ₙ C₂ₙ ⋯ Cₙₙ ⎠
Developer analogy: Before modifying module A, create a detailed document listing all the potential impacts on other modules B,C,D,…, and repeat this for every…