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Contoh Lengkap Determinan Matriks 5×5 (Ekspansi Kofaktor FULL)

Diberikan Matriks

A=(1234567891234567891234567)A = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 6 & 7 & 8 & 9 & 1 \\ 2 & 3 & 4 & 5 & 6 \\ 7 & 8 & 9 & 1 & 2 \\ 3 & 4 & 5 & 6 & 7 \end{pmatrix}
(1)

Langkah 1: Ekspansi Baris Pertama

det⁡(A)=1det⁡(A11)−2det⁡(A12)+3det⁡(A13)−4det⁡(A14)+5det⁡(A15)\det(A) = 1\det(A_{11}) -2\det(A_{12}) +3\det(A_{13}) -4\det(A_{14}) +5\det(A_{15})
(2)

Langkah 2: Minor 4×4

A11=(7891345689124567)A_{11} = \begin{pmatrix} 7 & 8 & 9 & 1 \\ 3 & 4 & 5 & 6 \\ 8 & 9 & 1 & 2 \\ 4 & 5 & 6 & 7 \end{pmatrix}
(3)
A12=(6891245679123567)A_{12} = \begin{pmatrix} 6 & 8 & 9 & 1 \\ 2 & 4 & 5 & 6 \\ 7 & 9 & 1 & 2 \\ 3 & 5 & 6 & 7 \end{pmatrix}
(4)
A13=(6791235678123467)A_{13} = \begin{pmatrix} 6 & 7 & 9 & 1 \\ 2 & 3 & 5 & 6 \\ 7 & 8 & 1 & 2 \\ 3 & 4 & 6 & 7 \end{pmatrix}
(5)
A14=(6781234678923457)A_{14} = \begin{pmatrix} 6 & 7 & 8 & 1 \\ 2 & 3 & 4 & 6 \\ 7 & 8 & 9 & 2 \\ 3 & 4 & 5 & 7 \end{pmatrix}
(6)
A15=(6789234578913456)A_{15} = \begin{pmatrix} 6 & 7 & 8 & 9 \\ 2 & 3 & 4 & 5 \\ 7 & 8 & 9 & 1 \\ 3 & 4 & 5 & 6 \end{pmatrix}
(7)

Langkah 3: Contoh Ekspansi 4×4 (DITULIS SATU PER SATU)

Ekspansi A11A_{11}

det⁡(A11)=7det⁡(B11)−8det⁡(B12)+9det⁡(B13)−1det⁡(B14)\det(A_{11}) = 7\det(B_{11}) -8\det(B_{12}) +9\det(B_{13}) -1\det(B_{14})
(8)

🔹 Minor 3×3 dari A11A_{11}

B11=(456912567)B_{11} = \begin{pmatrix} 4 & 5 & 6 \\ 9 & 1 & 2 \\ 5 & 6 & 7 \end{pmatrix}
(9)
det⁡(B11)=4(1⋅7−2⋅6)−5(9⋅7−2⋅5)+6(9⋅6−1⋅5)\det(B_{11}) = 4(1\cdot7 - 2\cdot6) -5(9\cdot7 - 2\cdot5) +6(9\cdot6 - 1\cdot5)
(10)
=4(7−12)−5(63−10)+6(54−5)=4(7-12)-5(63-10)+6(54-5)
(11)
=4(−5)−5(53)+6(49)=4(-5)-5(53)+6(49)
(12)
=−20−265+294=9=-20-265+294=9
(13)

Catatan

det⁡(A)=1det⁡(A11)−2det⁡(A12)+3det⁡(A13)−4det⁡(A14)+5det⁡(A15)\det(A) = 1\det(A_{11}) -2\det(A_{12}) +3\det(A_{13}) -4\det(A_{14}) +5\det(A_{15})
(14)
KOMPUTASI ALJABAR LINEAR
Rumus Determinan (Ekspansi Kofaktor)
KOMPUTASI ALJABAR LINEAR
Adjoint dan Invers Matriks 5x5 (Metode Kofaktor)