Invers matriks [1/(2(a-b)) 1/(2(a+b)) -1/(2(a-b))

Invers matriksnya adalah [[a-b, -(a-b)], [a+b, a+b]].

Pembahasan

Untuk mencari invers dari matriks yang diberikan, kita perlu menggunakan rumus invers matriks 2x2, yaitu:
Jika M = [[a, b], [c, d]], maka M⁻¹ = 1/(ad-bc) * [[d, -b], [-c, a]].

Matriks yang diberikan adalah:
M = [1/(2(a-b))   1/(2(a+b))]
    [-1/(2(a-b))  1/(2(a+b))]

Misalkan:
a' = 1/(2(a-b))
b' = 1/(2(a+b))
c' = -1/(2(a-b))
d' = 1/(2(a+b))

Maka, matriks tersebut adalah:
M = [[a', b'], [c', d']]

Pertama, kita hitung determinan matriks (ad - bc):
determinan = a'd' - b'c'
determinan = [1/(2(a-b))] * [1/(2(a+b))] - [1/(2(a+b))] * [-1/(2(a-b))]
determinan = 1/[4(a-b)(a+b)] + 1/[4(a+b)(a-b)]
determinan = 1/[4(a²-b²)] + 1/[4(a²-b²)]
determinan = 2/[4(a²-b²)]
determinan = 1/[2(a²-b²)]

Sekarang, kita gunakan rumus invers:
M⁻¹ = 1/determinan * [[d', -b'], [-c', a']]
M⁻¹ = 1 / (1/[2(a²-b²)]) * [[1/(2(a+b)), -1/(2(a+b))], [1/(2(a-b)), 1/(2(a-b))]]
M⁻¹ = 2(a²-b²) * [[1/(2(a+b)), -1/(2(a+b))], [1/(2(a-b)), 1/(2(a-b))]]

Mengalikan setiap elemen dengan 2(a²-b²):
M⁻¹ = [[2(a²-b²)/(2(a+b)), -2(a²-b²)/(2(a+b))], [2(a²-b²)/(2(a-b)), 2(a²-b²)/(2(a-b))]]
M⁻¹ = [[(a-b)(a+b)/(a+b), -(a-b)(a+b)/(a+b)], [(a-b)(a+b)/(a-b), (a-b)(a+b)/(a-b)]]
M⁻¹ = [[(a-b), -(a-b)], [(a+b), (a+b)]]

Jadi, invers matriks tersebut adalah [[a-b, -(a-b)], [a+b, a+b]].