Diketahui matriks A merupakan matriks singular berordo 2×2 yang memenuhi persamaan [(x,,1)] A [(x,,1)] = x² + 8x + 16. Jika matriks B merupakan matriks berordo 3 x 3 dengan B = [(1,,0)(0,,1)(1,,0)] A [(1,,0,,1)(0,,1,,0)], maka matriks B adalah ...

Jawaban : B = [(1 4 1)(4 16 4)(1 4 1)] Ingat! ▪️ Perkalian matriks A(n x m) dengan B(m x p) menghasilkan matriks ordo (n x p). Perkalian matriks dilakukan dengan mengalikan elemen baris dengan elemen kolom. Contoh perkalian matriks ordo 2 x 2 : [(a b)(c d)] [(e f)(g h)] = [(ae + bg af + bh)(ce + dg cf + dh)] ▪️ Determinan matriks [(a b)(c d)] adalah ad - bc ▪️ Matriks singular adalah matriks yang memiliki nilai determinan nol. Diketahui : A : matriks singular ordo (2x2) Misalkan A = [(a b)(c d)] det(A) = 0 ad - bc = 0 ...(i) [(x 1)][(a b)(c d)][(x)(1)] = x² + 8x + 16 [(x(a)+1(c) x(b)+1(d)][(x)(1)] = x² + 8x + 16 [(ax+c bx+d)][(x)(1)] = x² + 8x + 16 [((ax+c)(x)+(bx+d)(1))] = x² + 8x + 16 ax² +cx +bx + d = x² + 8x + 16 ax² + (c + b)x + d = x² + 8x + 16 Diperoleh a = 1 ....(ii) c + b = 8 ⇔ c = 8 - b ....(iii) d = 16 ...(iv) ad - bc = 0 1(16) - bc = 0 16 = bc 16 = b(8-b) 16 = 8b - b² b² - 8b + 16 = 0 (b - 4)² = 0 Nilai b = 4 ....(v) Substitusi b = 4 ke persamaan (iii) c = 8 - b c = 8 - 4 c = 4 diperoleh matriks A = [(1 4)(4 16)] B = [(1 0)(0 1)(1 0)]A[(1 0 1) (0 1 0)] = [(1 0)(0 1)(1 0)][(1 4)(4 16)][(1 0 1) (0 1 0)] = [(1(1)+0(4) 1(4)+0(16))(0(1)+1(4) 0(4)+1(16))(1(1)+0(4) 1(4)+0(16))][(1 0 1) (0 1 0)] = [(1 4)(4 16)(1 4)][(1 0 1) (0 1 0)] = [(1(1)+4(0) 1(0)+4(1) 1(1)+4(0)) (4(1)+16(0) 4(0)+16(1) 4(1)+16(0)) (1(1)+4(0) 1(0)+4(1) 1(1)+4(0))] = [(1 4 1)(4 16 4)(1 4 1)] Jadi, matriks B adalah B = [(1 4 1)(4 16 4)(1 4 1)]