Matriks Dan Ruang Vektor : Properti Dalam Determinan Dan Cramer's Rule ( Hukum Cramer )

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

http://informatika.unpar.ac.id/

Slide Materi MRV : Determinants in Matrices

Sumber http://wikiwoh.blogspot.com