Matriks Dan Ruang Vektor : Determinan Dan Perluasan Kofaktor Beserta Teladan Soal

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

If A is an n×n matrix, then

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]

Example 1 : 

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

Determinant of 2×2 matrices can be evaluated using

Example :

Determinant of 3×3 matrices

Determinant of 3×3 matrices which known as Sarrus Method : 

Example : 

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