Matriks dan Ruang Vektor : Properti dalam Determinan, Adjoint, dan Cramer's Rule ( Aturan Cramer )
Properties of Determinant
Suppose that A and B are n×n matrices and k is any scalar
- det(kA)=k^n det(A)
- det(A+B)≠det(A)+det(B)
- det(AB)=det(A) det(B)
Theorem :
A square matrix A is invertible if and only if det(A) ≠ 0
If A is invertible, then
Adjoint
If A is any n×n matrix and C_ij is the cofactor of a_ij, then the matrix
Is called the matrix of cofactors from A. The transpose of this matrix is called the adjoint of A ( denoted : adj(A) )
Example :
Note that :
The adjoint of A is
Inverse of a Matrix Using Its Adjoint
If A is an invertible matrix, then
Example :
det(A) = 64. Thus,
Cramer’s Rule
Use Cramer’s rule to solve
Equivalent Statement
Sumber
Slide Materi MRV : Determinants in Matrices
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