Minor, Cofactor, and Adjoin Matrix

If A is a rectangular matrix, the minor entries or elements aij are expressed by M_ij and are defined as the determinant of the submatrix that resides after the ith row and the jth column are crossed from A. The numbers (-1)^{i + j} Mij are expressed by C_ij called aij cofactor entries.

If A is any rectangular matrix (n x n) and C_ij is a cofactor a_ij, then the matrix:

is called the cofactor matrix of A. Transpose This matrix is called the adjoint of A and is denoted by Adj(A).

Example:
Determine the minor, cofactor, cofactor matrix, and adjoin of:

Answer:
Minor from matrix A is:
M₁₁ = 4
M₁₂ = 5
M₂₁ = 1
M₂₂ = -2

The cofactor of the matrix A is:
C₁₁ = (-1)¹⁺¹ M₁₁ = (1)4 = 4
C₁₂ = (-1)¹⁺² M₁₂ = (-1)5 = -5
C₂₁ = (-1)²⁺¹ M₂₁ = (-1)(1) = -1
C₂₂ = (-1)²⁺² M₂₂ = (1)(-2) = -2

The cofactor matrix is:

The adjoin of the cofactor matrix is the transpose of the cofactor matrix, so that:

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Referensi :

• To'Ali's book math group accounting and sales

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