Sebuah matriks disebut matriks ortogonal, jika Aᵗ=A⁻¹. Jika diketahui A=[(3/7 2/7 a)(b 3/7 2/7)(2/7 6/7 c)] adalah matriks ortogonal, maka a²+b²+c²=.... A. 81/49 B. 72/49 C. 45/49 D. 36/49 E. 9/49

Jawaban yang benar adalah A. Ingat! Perkalian matriks: A×B = [(a11 a12 a13)(a21 a22 a23)(a31 a32 a33)] × [(b11 b12 b13)(b21 b22 b23)(b31 b32 b33)] = [(a11×b11+a12×b21+a13×b31 a11×b12+a12×b22+a13×b32 a11×b13+a12×b23+a13×b33)(a21×b11+a22×b21+a23×b31 a21×b12+a22×b22+a23×b32 a21×b13+a22×b23+a23×b33)(a31×b11+a32×b21+a33×b31 a31×b12+a32×b22+a33×b32 a31×b13+a32×b23+a33×b33)] Transpose dari matriks A = [(a11 a12 a13)(a21 a22 a23)(a31 a32 a33)] adalah Aᵗ = [(a11 a21 a31)(a12 a22 a32)(a13 a23 a33)] Diketahui A = [(3/7 2/7 a)(b 3/7 2/7)(2/7 6/7 c)] Transpose dari matriks A adalah Aᵗ = [(3/7 b 2/7)(2/7 3/7 6/7)(a 2/7 c)] Jika Aᵗ = A⁻¹, maka A A⁻¹ = I A Aᵗ = I [(3/7 2/7 a)(b 3/7 2/7)(2/7 6/7 c)] [(3/7 b 2/7)(2/7 3/7 6/7)(a 2/7 c)] = [(1 0 0)(0 1 0)(0 0 1)] [(3/7·3/7+2/7·2/7+a·a 3/7·b+2/7·3/7+a·2/7 3/7·2/7+2/7·6/7+a·c)(b·3/7+3/7·2/7+2/7·a b·b+3/7·3/7+2/7·2/7 b·2/7+3/7·6/7+2/7·c)(2/7·3/7+6/7·2/7+c·a 2/7·b+6/7·3/7+c·2/7 2/7·2/7+6/7·6/7+c·c)] = [(1 0 0)(0 1 0)(0 0 1)] [(9/49+4/49+a² 3/7·b+6/49+2/7·a 6/49+12/49+a·c)(3/7·b+6/49+2/7·a b²+9/49+4/49 2/7·b+18/49+2/7·c)(6/49+12/49+c·a 2/7·b+18/49+2/7·c 4/49+36/49+c²)] = [(1 0 0)(0 1 0)(0 0 1)] [(a²+13/49 2/7·a+3/7·b+6/49 a·c+18/49)(2/7·a+3/7·b+6/49 b²+13/49 2/7·b+2/7·c+18/49)(a·c+18/49 2/7·b+2/7·c+18/49 c²+40/49)] = [(1 0 0)(0 1 0)(0 0 1)] Diperoleh a²+13/49 = 1 2/7·a+3/7·b+6/49 = 0 a·c+18/49 = 0 2/7·a+3/7·b+6/49 =0 b²+13/49 = 1 2/7·b+2/7·c+18/49 = 0 a·c+18/49 = 0 2/7·b+2/7·c+18/49 = 0 c²+40/49 = 1 Perhatikan a²+13/49 = 1 a² = 1-13/49 a² = 36/49 b²+13/49 = 1 b² = 1-13/49 b² = 36/49 c²+40/49 = 1 c² = 1-40/49 c² = 9/49 Sehingga a²+b²+c² = 36/49+36/49+9/49 = 81/49 Jadi, a²+b²+c² = 81/49. Pilihan jawaban yang benar adalah A.