Matriks dan Ruang Vektor : Determinan dan Ekspansi Kofaktor beserta Contoh Soal
Determinant
The 2×2 matrix of
is invertible if and only if ad - bc ≠ 0
The expression ad-bc is a determinant of the matrix A
det(A) = ad - bc
or
After finds out the value of certain determinant of matrices, we could use the determinant for finding the inverse matrices as shown syntax below :
Example :
If
then det(A) =…
Minor and Cofactor
A is a square matrix. The minor of entry a_ij is denoted by M_ij. M_ij is a determinant of the submatrix that remains after the ith row and j-th column are deleted from A.
The number (-1)^(i+j) M_ij is denoted by C_ij (cofactor of entry a_ij)
Example :
Let
The minor of entry a_11 is
The cofactor of a_11 is
Please note that a minor M_ij and its corresponding cofactor C_ij are either the same or negatives of each other.
Example :
Cofactor expansions of a 2×2 matrix, the checkerboard pattern :
so that concluded
Determinant by Cofactor Expansion
det(A) = a_1jC_1j + a_2jC_2j + … + a_njC_nj, for j ∈ [1,n]
Or
det(A) = a_i1C_i1 + a_i2C_i2 + … + a_inC_in, for i ∈ [1,n]
Find the determinant of the matrix by cofactor expansion along the first row !
Example 2 :
If A is the 4×4 matrix
Find det(A) !
Determinant of a Lower Triangular Matrix
Notice that determinant of a 4×4 lower triangular matrix is the product of its diagonal entries
Theorem :
If A is an n×n triangular matrix (upper, lower, or diagonal), then det〖(A)〗 is the product of the entries on the main diagonal of the matrix
det(A) = a_11a_22 …a_nn
Determinant of 2×2
Example :
Determinant of 3×3 matrices
Example :
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