MATRIKS Fakultas Kehutanan Universitas Lambung Mangkurat Matematika II - 22 Matriks

MATRIKS * M b x l

Bentuk Matriks Matriks Segi Matriks Tak Segi (m = n) b11 b12 b13 ………… b1n b21 b22 b23 ………… b2n b31 b32 b33 ………… b3n . . . . . . . . . bm1 bm2 bm3 ……….. bmn

M3 = A4 = S4 = M4 = Matriks Segi Matriks setangkup 9 4 6 0 8 5 2 8 2 -1 3 1 3 4 0 0 9 5 2 7 8 1 4 -6 Matriks setangkup Matriks miring setangkup S4 = M4 = 2 3 2 5 3 9 0 1 9 0 6 4 5 1 4 7 0 -1 2 -2 1 0 6 8 -2 -6 0 5 2 -8 -5 0

Matriks segitiga bawah Matriks diagonal Matriks tanda D4 = T3 = 1 0 0 0 -1 0 0 0 -1 2 0 0 0 0 -8 0 0 0 0 1 0 0 0 0 5 Matriks segitiga atas Matriks segitiga bawah A4 = 1 4 2 1 0 3 7 2 0 0 2 4 0 0 0 9 B4 = 1 0 0 0 4 3 0 0 2 7 2 0 1 2 4 9

Matriks nol Matriks satu N3 = S3 = Matriks satu-nol M3 = 0 0 0 S3 = 1 1 1 Matriks satu-nol M3 = 1 0 1 0 1 1 0 1 0 Matriks skalar Matriks Identitas S4 = I4 = 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

Matriks Tak Segi Matriks datar Matriks tegak W = M = Matriks nol X = 1 6 5 2 9 0 2 4 4 8 8 7 9 4 0 8 5 2 8 2 4 7 Matriks nol X = 4 x 3 0 0 0 E = 2 x 3 0 0 0

M = S = F = Y = Matriks satu Matriks satu-nol 4 x 2 1 1 1 3 x 4 1 1 S = 3 x 4 1 1 1 1 Matriks satu-nol F = 4 x 3 0 1 0 1 1 1 0 0 1 1 1 Y = 3 x 4 0 0 1 1 1 0 1 0 1 1 0 1

Penjumlahan 2 Matriks B ba x la bb x lb ( ba = bb & la = lb ) Hanya berlaku bila : A ba x la B bb x lb ( ba = bb & la = lb ) A = 4 x 3 a11 a12 a13 a21 a22 a23 a31 a32 a33 a41 a42 a43 B = 4 x 3 b11 b12 b13 b21 b22 b23 b31 b32 b33 b41 b42 b43

* Tambah A + B = A + B = = a11+b11 a12+b12 a13+b13 (4 x 3) a11+b11 a12+b12 a13+b13 a21+b21 a22+b22 a23+b23 a31+b31 a32+b32 a33+b33 a41+b41 a42+b42 a43+b43 A = 2 x 3 2 3 5 3 0 -1 B = 2 x 3 3 6 1 2 0 1 A + B = = (2 x 3) 2+3 3+6 5+1 3+2 0+0 -1+1 5 9 6 5 0 0

* Kurang A - B = A - B = = a11-b11 a12-b12 a13-b13 (4 x 3) a11-b11 a12-b12 a13-b13 a21-b21 a22-b22 a23-b23 a31-b31 a32-b32 a33-b33 a41-b41 a42-b42 a43-b43 A = 2 x 3 2 3 5 3 0 -1 B = 2 x 3 3 6 1 2 0 1 A - B = = (2 x 3) 2-3 3-6 5-1 3-2 0-0 -1-1 -1 -3 4 1 0 -2

Penggandaan 2 Matriks A x B = C ba x la bb x lb ba x lb ( bb = la) Hanya berlaku bila : A x B = C ba x la bb x lb ba x lb ( bb = la) A = 4 x 2 a11 a12 a21 a22 a31 a32 a41 a42 B = 2 x 3 b11 b12 b13 b21 b22 b23

A x B = C = a11b11+a12b21 a11b12+a12b22 a11b13+a12b23 c11+c11 c12+c12 c13+c13 c21+c21 c22+c22 c23+c23 c31+c31 c32+c32 c33+c33 c41+c41 c42+c42 c43+c43

(2)(3)+(3)(6)+(5)(1) (2)(2)+(3)(0)+(5)(1) 3 2 6 0 1 1 A = 2 x 3 2 3 5 3 0 -1 B = 3 x 2 3 2 6 0 1 1 A x B = 2 x 3 3 x 2 2 3 5 3 0 -1 = (2)(3)+(3)(6)+(5)(1) (2)(2)+(3)(0)+(5)(1) (3)(3)+(0)(6)+(-1)(1) (3)(2)+(0)(0)+(-1)(1) = 9 8 5

(3)(2)+(2)(3) (3)(3)+(2)(0) (3)(5)+(2)(-1) 3 2 6 0 1 1 B x A = 3 x 2 2 x 3 2 3 5 3 0 -1 = (3)(2)+(2)(3) (3)(3)+(2)(0) (3)(5)+(2)(-1) (6)(2)+(0)(3) (6)(3)+(0)(0) (6)(5)+(0)(-1) (1)(2)+(1)(3) (1)(3)+(1)(0) (1)(5)+(1)(-1) = 12 9 13 12 18 30 5 3 4

M = (mij)bl M’ = (m’ji)lb Putaran Suatu Matriks M = (mij)bl M’ = (m’ji)lb  M = 3 x 2 m11 m12 m21 m22 m31 m32  m11 m21 m31 m12 m22 m32 M’ = 2 x 3  M = 3 x 4 4 1 4 0 0 7 4 9 5 1 M = 4 x 3 2 3 4 0 9 1 0 5 4 7 1

Teras Suatu Matriks M = (mij)bb tr M = mii = m11 + m22 + m33 ……….. + mbb M3 = m11 m12 m13 m21 m22 m23 m31 m32 m33 tr M = m11 + m22 + m33 M3 = 2 5 0 3 6 9 6 1 4 tr M = 2 + 6 + 4 = 12

Matriks Sekatan Pengolahan ganda pada 2 buah matriks yang berdimensi (ukuran) besar biasanya sulit dilakukan. Untuk memudahkannya dilakukan penyekatan sehingga terbentuk anak-anak matriks dengan dimensi yang lebih kecil. Cara penyekat harus memperhatikan ketentuan bahwa banyak jalur pada anak-matriks yang digandakan harus samadengan banyaknya baris anak-matriks pengganda.

M = = m11 m12 m13 m14 ………… b1l m21 m22 m23 m24 ………… b2l . . . . . . . . . . . mb1 mb2 mb3 mb4 ……….. mbl M = b x l = M11 M12 (p x q) p(l – q) M21 M22 (b – p)q (b – p)(l – q)

N = = n11 n12 n13 ………… n1k n21 n22 n23 ………… n2k n31 n32 n33 ………… n3k . . . . nl1 nl2 nl3 ………... nlk N = l x k N11 N12 (q x r) q(k – r) N21 N22 (l – q)r (l – q)(k – r) =

M x N = C C = M11 N11 + M12 N21 M11 N12 + M12 N22 (p x r) p(k-r) b x l l x k b x k M11 N11 + M12 N21 M11 N12 + M12 N22 (p x r) p(k-r) M21 N11 + M22 N21 M21 N12 + M22 N22 (b – p)r (b – p)(k – r) C = b x k

Transformasi Dasar a11 a12 a13 ………… a1n A = a21 a22 a23 ………… a2n (pengolahan baris atau lajur terhadap suatu matriks dengan cara pertukaran letak, penjumlahan atau penggandaan) a11 a12 a13 ………… a1n a21 a22 a23 ………… a2n a31 a32 a33 ………… a3n . . . . am1 am2 am3 ……….. amn A =

Pertukaran letak E1.2 F1.2 A A 1 2 1 3 4 2 4 6 x = 2 3 1 3 4 2 1 2 1 2 1 3 4 2 4 6 x = 2 3 Pertukaran letak 1 3 4 2 1 2 2 4 6 1 2 2 3 1 4 4 2 6 E1.2 F1.2 A A

Penjumlahan E3.2(1) A Tambah F3.2(1) A 1 2 1 3 4 3 7 10 Brs 3 : 2 4 6 1 2 1 3 4 3 7 10 Brs 3 : 2 4 6 Brs 2 x 1 : 1 3 4 + 3 7 10 E3.2(1) A Tambah 2 4 6 1 3 + 7 10 Ljr 2 x 1 Ljr 3 1 3 1 3 7 2 4 10 F3.2(1) A

E3.2(-1) A Kurang F3.2(-1) A 2 1 2 1 3 4 1 1 2 Brs 3 : 2 4 6 2 1 2 1 3 4 1 1 2 Brs 3 : 2 4 6 Brs 2 x (-1) : -1 -3 -4 + 1 1 2 E3.2(-1) A Kurang 2 4 6 -1 -3 -4 1 + Ljr 3 Ljr 2 x (-1) 1 1 1 3 1 2 4 2 F3.2(-1) A

Penggandaan K a l i E3(2) F3(2) A A B a g i E3(1/2) F3(1/2) A A 1 4 1 2 1 3 4 4 8 12 1 4 1 3 8 2 4 12 E3(2) F3(2) A A B a g i 1 2 1 3 4 1 2 3 1 1 1 3 2 2 4 3 E3(1/2) F3(1/2) A A