Bab 15 Matriks

Posted on May 7, 2016 by user

4.8 Penyelesaian Persamaan Linear Serentak dengan Kaedah Matriks

1. Dua persamaan linear serentak boleh ditulis dalam bentuk persamaan matriks.

Sebagai contoh, dalam persamaan linear serentak:

ax+ by = c

dx+ ey = f

boleh ditulis dalam format persamaan matriks seperti berikut:

(\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{array}{l} x \\ y \end{array}) = (\begin{array}{l} e \\ f \end{array}),

Di mana a, b, c, d, e dan fadalah pemalar manakala x dan y adalah anu.

Contoh 1:

Tuliskan persamaan linear serentak berikut dalam bentuk persamaan matriks.

y– 6x – 19 = 0

2y + 3x + 22 = 0

Penyelesaian:

– 6x + y = 19

3x + 2y = – 22

Persamaan matriks ialah:

(\begin{matrix} - 6 & 1 \\ 3 & 2 \end{matrix}) (\begin{array}{l} x \\ y \end{array}) = (\begin{array}{l} 19 \\ - 22 \end{array})

2. Persamaan matriks dalam bentuk

(\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{array}{l} x \\ y \end{array}) = (\begin{array}{l} e \\ f \end{array})

dapat diselesaikan bagi anu x dan y seperti berikut.

(a) Katakan

A = (\begin{matrix} a & b \\ c & d \end{matrix})

dan cari A^-1.

(b) Darabkan kedua-dua belah persamaan dengan A^-1.

A^{- 1} (\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{array}{l} x \\ y \end{array}) = A^{- 1} (\begin{array}{l} e \\ f \end{array})

\begin{array}{l} (c) A^{- 1} A (\begin{array}{l} x \\ y \end{array}) = A^{- 1} (\begin{array}{l} e \\ f \end{array}) \\ I (\begin{array}{l} x \\ y \end{array}) = A^{- 1} (\begin{array}{l} e \\ f \end{array}) \\ ↗ \\ A^{- 1} A = I = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) \\ (\begin{array}{l} x \\ y \end{array}) = A^{- 1} (\begin{array}{l} e \\ f \end{array}) \\ (\begin{array}{l} x \\ y \end{array}) = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix}) (\begin{array}{l} e \\ f \end{array}) \end{array}

Contoh 2:

Selesaikan persamaan linear serentak berikut dalam bentuk persamaan matriks.

2x = 5 – 3y

7x = 1 – 5y

Penyelesaian:

2x + 3y = 5

7x + 5y = 1

(\begin{matrix} 2 & 3 \\ 7 & 5 \end{matrix}) (\begin{array}{l} x \\ y \end{array}) = (\begin{array}{l} 5 \\ 1 \end{array}) \leftarrow \begin{array}{l} Tulis persamaan serentak \\ dalam bentuk matriks . \end{array}

\begin{array}{l} Katakan A = (\begin{matrix} 2 & 3 \\ 7 & 5 \end{matrix}) \\ A^{- 1} = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix}) \\ A^{- 1} = \frac{1}{10 - 21} (\begin{matrix} 5 & - 3 \\ - 7 & 2 \end{matrix}) \\ A^{- 1} = \frac{1}{- 11} (\begin{matrix} 5 & - 3 \\ - 7 & 2 \end{matrix}) \end{array}

(\begin{array}{l} x \\ y \end{array}) = \frac{1}{- 11} (\begin{matrix} 5 & - 3 \\ - 7 & 2 \end{matrix}) (\begin{array}{l} 5 \\ 1 \end{array}) \leftarrow (\begin{array}{l} x \\ y \end{array}) = A^{- 1} (\begin{array}{l} e \\ f \end{array})

\begin{array}{l} (\begin{array}{l} x \\ y \end{array}) = \frac{1}{- 11} (\begin{array}{l} 5 \times 5 + (- 3) \times 1 \\ - 7 \times 5 + 2 \times 1 \end{array}) \\ (\begin{array}{l} x \\ y \end{array}) = \frac{1}{- 11} (\begin{array}{l} 22 \\ - 33 \end{array}) \\ (\begin{array}{l} x \\ y \end{array}) = (\begin{array}{l} - \frac{22}{11} \\ \frac{- 33}{- 11} \end{array}) = (\begin{array}{l} - 2 \\ 3 \end{array}) \\ ∴ x = - 2, y = 3. \end{array}

Bab 15 Matriks

Posted on May 6, 2016 by user

4.5 Pendaraban Dua Matriks (Contoh Soalan)

Soalan 1:

Cari hasil darab bagi setiap pasangan matriks yang berikut.

\begin{array}{l} (a) (1 5 2) (\begin{array}{l} 2 \\ 4 \\ 3 \end{array}) \\ (b) (\begin{matrix} 2 & 8 \\ - 3 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ 4 & - 2 \end{matrix}) \end{array}

\begin{array}{l} (c) (\begin{array}{l} - 3 \\ 5 \end{array}) (2 6) \\ (d) (\begin{matrix} 0 & 4 \\ - 1 & 3 \end{matrix}) (\begin{array}{l} 7 \\ - 2 \end{array}) \\ (e) (7 - 4) (\begin{matrix} - 2 & 0 \\ - 1 & 3 \end{matrix}) \end{array}

Penyelesaian:

\begin{array}{l} (a) \\ (1 5 2) (\begin{array}{l} 2 \\ 4 \\ 3 \end{array}) \leftarrow \begin{array}{l} Analisis matriks \\ 1 \times 3 dan 3 \times 1 \\ ↘ ↙ \\ = matriks 1 \times 1 \end{array} \\ = (1 \times 2 \oplus 5 \times 4 \oplus 2 \times 3) \\ = (2 + 20 + 6) \\ = (28) \end{array}

\begin{array}{l} (b) \\ (\begin{matrix} 2 & 8 \\ - 3 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ 4 & - 2 \end{matrix}) \leftarrow \begin{array}{l} Analisis matriks \\ 2 \times 2 dan 2 \times 2 \\ ↘ ↙ \\ = matriks 2 \times 2 \end{array} \\ = (\begin{matrix} 2 \times 1 + 8 \times 4 & 2 \times 0 + 8 \times - 2 \\ - 3 \times 1 + 1 \times 4 & - 3 \times 0 + 1 \times - 2 \end{matrix}) \\ = (\begin{matrix} 34 & - 16 \\ 1 & - 2 \end{matrix}) \end{array}

\begin{array}{l} (c) \\ (\begin{array}{l} - 3 \\ 5 \end{array}) (2 6) \leftarrow \begin{array}{l} Analisis matriks \\ 2 \times 1 dan 1 \times 2 \\ ↘ ↙ \\ = matriks 2 \times 2 \end{array} \\ = (\begin{matrix} - 3 \times 2 & - 3 \times 6 \\ 5 \times 2 & 5 \times 6 \end{matrix}) \\ = (\begin{matrix} - 6 & - 18 \\ 10 & 30 \end{matrix}) \end{array}

\begin{array}{l} (d) \\ (\begin{matrix} 0 & 4 \\ - 1 & 3 \end{matrix}) (\begin{array}{l} 7 \\ - 2 \end{array}) \leftarrow \begin{array}{l} Analisis matriks \\ 2 \times 2 dan 2 \times 1 \\ ↘ ↙ \\ = matriks 2 \times 1 \end{array} \\ = (\begin{array}{l} 0 \times 7 + 4 \times - 2 \\ - 1 \times 7 + 3 \times - 2 \end{array}) \\ = (\begin{array}{l} - 8 \\ - 13 \end{array}) \end{array}

\begin{array}{l} (e) \\ (7 - 4) (\begin{matrix} - 2 & 0 \\ - 1 & 3 \end{matrix}) \leftarrow \begin{array}{l} Analisis matriks \\ 1 \times 2 dan 2 \times 2 \\ ↘ ↙ \\ = matriks 1 \times 2 \end{array} \\ = (7 \times - 2 + (- 4 \times - 1) 7 \times 0 + (- 4 \times 3)) \\ = (- 14 + 4 0 - 12) \\ = (- 10 - 12) \end{array}

Contoh 2:

Cari nilai mdan nilai n dalam setiap persamaan matriks yang berikut:

(a) (\begin{array}{l} 3 \\ m \end{array}) (1 n) = (\begin{matrix} 3 & 12 \\ - 2 & - 8 \end{matrix})

(b) (\begin{matrix} m & 2 \\ - 3 & 1 \end{matrix}) (\begin{array}{l} 2 \\ n \end{array}) = (\begin{array}{l} 12 \\ 4 + 2 n \end{array})

(c) (\begin{matrix} m & - 3 \\ - 1 & 1 \end{matrix}) (\begin{matrix} - 1 & 2 \\ 4 & n \end{matrix}) = (\begin{matrix} - 14 & - 11 \\ 5 & 3 \end{matrix})

Penyelesaian:

\begin{array}{l} (a) \\ (\begin{array}{l} 3 \\ m \end{array}) (1 4) = (\begin{matrix} 3 & 12 \\ - 2 & n \end{matrix}) \\ (\begin{matrix} 3 & 12 \\ m & 4 m \end{matrix}) = (\begin{matrix} 3 & 12 \\ - 2 & n \end{matrix}) \end{array}

m= –2,

4m = n

4 (–2) = n

n= –8

\begin{array}{l} (b) \\ (\begin{matrix} m & 2 \\ - 3 & 1 \end{matrix}) (\begin{array}{l} 2 \\ n \end{array}) = (\begin{array}{l} 12 \\ 4 + 2 n \end{array}) \\ (\begin{array}{l} 2 m + 2 n \\ - 6 + n \end{array}) = (\begin{array}{l} 12 \\ 4 + 2 n \end{array}) \end{array}

–6 + n = 4 + 2n

n= –10

2m + 2n = 12

2m + 2 (–10) = 12

2m – 20 = 12

2m = 32

m = 16

\begin{array}{l} (c) \\ (\begin{matrix} m & - 3 \\ - 1 & 1 \end{matrix}) (\begin{matrix} - 1 & 2 \\ 4 & n \end{matrix}) = (\begin{matrix} - 14 & - 11 \\ 5 & 3 \end{matrix}) \\ (\begin{matrix} - m + (- 12) & 2 m + (- 3 n) \\ 1 + 4 & - 2 + n \end{matrix}) = (\begin{matrix} - 14 & - 11 \\ 5 & 3 \end{matrix}) \end{array}

–m – 12 = –14

–m = –2

m = 2

–2 + n = 3

n = 5

Bab 15 Matriks

Posted on May 6, 2016 by user

4.5 Pendaraban Dua Matriks

1. Dua matriks hanya boleh didarab apabila bilangan lajur matriks pertama sama dengan bilangan baris matriks kedua.

2. Contohnya, jika Aialah suatu matriks m × n dan Bialah suatu matriks n × t, maka hasil darab matriks AB = P. P ialah suatu matriks, m× t.

Contoh:

\begin{array}{l} (a) (a b) (\begin{array}{l} c \\ d \end{array}) = (a c + b d) \\ 1 \times 22 \times 1 1 \times 1 \\ (b) (\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{array}{l} e \\ f \end{array}) = (\begin{array}{l} a e + b f \\ c e + d f \end{array}) \\ 2 \times 22 \times 12 \times 1 \end{array}

\begin{array}{l} (c) (\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{matrix} e & f \\ g & h \end{matrix}) = (\begin{matrix} a e + b g & a f + b h \\ c e + d g & c f + d h \end{matrix}) \\ 2 \times 22 \times 22 \times 2 \\ (d) (\begin{array}{l} a \\ b \end{array}) (c d) = (\begin{matrix} a c & a d \\ b c & b d \end{matrix}) \\ 2 \times 11 \times 22 \times 2 \end{array}

\begin{array}{l} (e) (a b c) (\begin{array}{l} d \\ e \\ f \end{array}) = (a d + b e + c f) \\ 1 \times 33 \times 11 \times 1 \\ (f) (\begin{matrix} a & b \\ \begin{array}{l} c \\ e \end{array} & \begin{array}{l} d \\ f \end{array} \end{matrix}) (\begin{array}{l} g \\ h \end{array}) = (\begin{array}{l} a g + b h \\ c g + d h \\ e g + f h \end{array}) \\ 3 \times 22 \times 13 \times 1 \end{array}

Contoh 1:

Tentukan sama ada hasil darab matriks yang berikut boleh dialakukan atau tidak. Jika boleh, nyatakan peringkat matriks yang terhasil.

\begin{array}{l} (a) (\begin{matrix} 3 & - 5 \\ 1 & 2 \end{matrix}) (3 7) \\ (b) (\begin{matrix} 2 & 9 \\ 1 & 3 \end{matrix}) (\begin{array}{l} 8 \\ 6 \end{array}) \\ (c) (- 10 - 6) (\begin{array}{l} 7 \\ 2 \end{array}) \\ (d) (\begin{array}{l} 8 \\ 6 \end{array}) (\begin{matrix} 2 & 9 \\ 1 & 3 \end{matrix}) \\ (e) (\begin{array}{l} 7 \\ - 3 \end{array}) (2 10) \end{array}

Penyelesaian:

\begin{array}{l} (a) (\begin{matrix} 3 & - 5 \\ 1 & 2 \end{matrix}) (3 7) \\ 2 \times 2 1 \times 2 \leftarrow 2 \neq 1 ∴ Tidak boleh didarab . \end{array}

\begin{array}{l} b) (\begin{matrix} 2 & 9 \\ 1 & 3 \end{matrix}) (\begin{array}{l} 8 \\ 6 \end{array}) \\ 2 \times 2 2 \times 1 \leftarrow 2 = 2 ∴ Boleh didarab . \\ Peringkat matriks yang terhasil = 2 \times 1 \end{array}

\begin{array}{l} (c) (- 10 - 6) (\begin{array}{l} 7 \\ 2 \end{array}) \\ 1 \times 2 2 \times 1 \leftarrow 2 = 2 ∴ Boleh didarab . \\ Peringkat matriks yang terhasil = 1 \times 1 \end{array}

\begin{array}{l} (d) (\begin{array}{l} 8 \\ 6 \end{array}) (\begin{matrix} 2 & 9 \\ 1 & 3 \end{matrix}) \\ 2 \times 1 2 \times 2 \leftarrow 1 \neq 2 ∴ Tidak boleh didarab . \end{array}

\begin{array}{l} (e) (\begin{array}{l} 7 \\ - 3 \end{array}) (2 10) \\ 2 \times 1 1 \times 2 \leftarrow 1 = 1 ∴ Boleh didarab . \\ Peringkat matriks yang terhasil = 2 \times 2 \end{array}

Bab 15 Matriks

Posted on May 6, 2016 by user

4.4 Pendaraban Matriks dengan Nombor (Contoh Soalan)

Contoh 1:

Ungkapkan setiap yang berikut sebagai satu matriks tunggal.

(a) 3 (\begin{matrix} 1 & - 4 \\ - 3 & 2 \end{matrix}) + 2 (\begin{matrix} - 3 & 1 \\ 3 & - 4 \end{matrix})

(b) 2 (\begin{matrix} - 1 & 0 \\ 9 & - 4 \end{matrix}) - 4 (\begin{matrix} - 2 & 2 \\ 3 & - 6 \end{matrix}) - \frac{1}{3} (\begin{matrix} - 9 & - 3 \\ - 6 & 15 \end{matrix})

Penyelesaian:

\begin{array}{l} (a) 3 (\begin{matrix} 1 & - 4 \\ - 3 & 2 \end{matrix}) + 2 (\begin{matrix} - 3 & 1 \\ 3 & - 4 \end{matrix}) \\ = (\begin{matrix} 3 \times 1 & 3 \times (- 4) \\ 3 \times (- 3) & 3 \times 2 \end{matrix}) + (\begin{matrix} 2 \times (- 3) & 2 \times 1 \\ 2 \times 3 & 2 \times (- 4) \end{matrix}) \\ = (\begin{matrix} 3 & - 12 \\ - 9 & 6 \end{matrix}) + (\begin{matrix} - 6 & 2 \\ 6 & - 8 \end{matrix}) \\ = (\begin{matrix} 3 + (- 6) & - 12 + 2 \\ - 9 + 6 & 6 - (- 8) \end{matrix}) \\ = (\begin{matrix} - 3 & - 10 \\ - 3 & 14 \end{matrix}) \end{array}

\begin{array}{l} (b) 2 (\begin{matrix} - 1 & 0 \\ 9 & - 4 \end{matrix}) - 4 (\begin{matrix} - 2 & 2 \\ 3 & - 6 \end{matrix}) - \frac{1}{3} (\begin{matrix} - 9 & - 3 \\ - 6 & 15 \end{matrix}) \\ = (\begin{matrix} - 2 & 0 \\ 18 & - 8 \end{matrix}) - (\begin{matrix} - 8 & 8 \\ 12 & - 24 \end{matrix}) - (\begin{matrix} \frac{1}{3} \times (- 9) & \frac{1}{3} \times (- 3) \\ \frac{1}{3} \times (- 6) & \frac{1}{3} \times 15 \end{matrix}) \\ = (\begin{matrix} - 2 & 0 \\ 18 & - 8 \end{matrix}) - (\begin{matrix} - 8 & 8 \\ 12 & - 24 \end{matrix}) - (\begin{matrix} - 3 & - 1 \\ - 2 & 5 \end{matrix}) \\ = (\begin{matrix} - 2 - (- 8) - (- 3) & 0 - 8 - (- 1) \\ 18 - 12 - (- 2) & - 8 - (- 24) - 5 \end{matrix}) \\ = (\begin{matrix} 9 & - 7 \\ 10 & 11 \end{matrix}) \end{array}

Bab 15 Matriks

Posted on May 6, 2016 by user

4.4 Pendaraban Matriks dengan Nombor

Pendaraban suatu matriks dengan suatu nombor ialah pendaraban setiap unsur matriks dengan nombor itu.

Contoh:

Diberi

A = (\begin{matrix} - 2 & 4 \\ 5 & - 6 \end{matrix})

, cari setiap yang berikut sebagai satu matriks tunggal.

(a) 3A

(b) -2A

Penyelesaian:

\begin{array}{l} (a) 3 A = 3 (\begin{matrix} - 2 & 4 \\ 5 & - 6 \end{matrix}) \\ = (\begin{matrix} 3 \times (- 2) & 3 \times 4 \\ 3 \times 5 & 3 \times (- 6) \end{matrix}) \\ = (\begin{matrix} - 6 & 12 \\ 15 & - 18 \end{matrix}) \end{array}

\begin{array}{l} (b) - 2 A = - 2 (\begin{matrix} - 2 & 4 \\ 5 & - 6 \end{matrix}) \\ = (\begin{matrix} - 2 \times (- 2) & - 2 \times 4 \\ - 2 \times 5 & - 2 \times (- 6) \end{matrix}) \\ = (\begin{matrix} - 4 & - 8 \\ - 10 & 12 \end{matrix}) \end{array}

Bab 15 Matriks

Posted on May 6, 2016 by user

4.3 Penambahan dan Penolakan Matriks (Contoh Soalan)

Contoh 1:

Cari hasil tambah bagi matriks yang berikut:

\begin{array}{l} (a) (18 - 7) + (3 6) \\ (b) (\begin{matrix} - 13 & 0 \\ 7 & - 1 \end{matrix}) + (\begin{matrix} - 1 & 3 \\ 5 & 6 \end{matrix}) \end{array}

Penyelesaian:

\begin{array}{l} (a) (18 - 7) + (3 6) \\ = (18 + 3 - 7 + 6) = (21 - 1) \\ (b) (\begin{matrix} - 13 & 0 \\ 7 & - 1 \end{matrix}) + (\begin{matrix} - 1 & 3 \\ 5 & 6 \end{matrix}) \\ = (\begin{matrix} - 13 + (- 1) & 0 + 3 \\ 7 + 5 & - 1 + 6 \end{matrix}) = (\begin{matrix} - 14 & 3 \\ 12 & 5 \end{matrix}) \end{array}

Contoh 2:

Cari setiap yang berikut sebagai satu matriks tunggal.

\begin{array}{l} (a) (\begin{array}{l} 9 \\ 6 \end{array}) - (\begin{array}{l} 7 \\ - 2 \end{array}) \\ (b) (\begin{matrix} 3 & - 4 \\ 0 & 5 \end{matrix}) - (\begin{matrix} - 7 & - 3 \\ - 6 & 1 \end{matrix}) \end{array}

Penyelesaian:

\begin{array}{l} (a) (\begin{array}{l} 9 \\ 6 \end{array}) - (\begin{array}{l} 7 \\ - 2 \end{array}) = (\begin{array}{l} 9 - 7 \\ 6 - (- 2) \end{array}) = (\begin{array}{l} 2 \\ 8 \end{array}) \\ (b) (\begin{matrix} 3 & - 4 \\ 0 & 5 \end{matrix}) - (\begin{matrix} - 7 & - 3 \\ - 6 & 1 \end{matrix}) \\ = (\begin{matrix} 3 - (- 7) & - 4 - (- 3) \\ 0 - (- 6) & 5 - 1 \end{matrix}) \\ = (\begin{matrix} 3 + 7 & - 4 + 3 \\ 0 + 6 & 5 - 1 \end{matrix}) = (\begin{matrix} 10 & - 1 \\ 6 & 4 \end{matrix}) \end{array}

(B) Menentukan nilai unsur yang tidak diketahui dalam persamaan matriks yang melibatkan operasi tambah dan tolak

Contoh 3:

Diberi

(\begin{array}{l} p \\ q \end{array}) + (\begin{array}{l} 5 p \\ 9 \end{array}) = (\begin{array}{l} - 12 \\ 3 q + 1 \end{array})

, cari nilai bagi p dan q.

Penyelesaian:

\begin{array}{l} (\begin{array}{l} p \\ q \end{array}) + (\begin{array}{l} 5 p \\ 9 \end{array}) = (\begin{array}{l} - 12 \\ 3 q + 1 \end{array}) \\ (\begin{array}{l} p + 5 p \\ q + 9 \end{array}) = (\begin{array}{l} - 12 \\ 3 q + 1 \end{array}) \\ p + 5 p = - 12 \\ 6 p = - 12 \\ p = - 2 \end{array}

\begin{array}{l} q + 9 = 3 q + 1 \\ q - 3 q = 1 - 9 \\ - 2 q = - 8 \\ q = 4 \end{array}

Contoh 4:

Cari nilai m dan n bagi persamaan matriks yang berikut:

(\begin{matrix} 7 & 6 \\ n & 1 \end{matrix}) - (\begin{matrix} - 1 & m \\ - 4 & - 2 \end{matrix}) = (\begin{matrix} 5 & 12 \\ 6 & 11 \end{matrix})

Penyelesaian:

\begin{array}{l} (\begin{matrix} 7 & 6 \\ n & 1 \end{matrix}) - (\begin{matrix} - 1 & m \\ - 4 & - 2 \end{matrix}) = (\begin{matrix} 5 & 12 \\ 6 & 11 \end{matrix}) \\ 6 - m = 12 \\ - m = 6 \\ m = - 6 \\ n - (- 4) = 6 \\ n + 4 = 6 \\ n = 2 \end{array}

Bab 15 Matriks

Posted on May 6, 2016 by user

4.2 Matriks Sama (Contoh Soalan)

Contoh 1:

(\begin{matrix} 1 & x + 2 \\ 4 - y & - 1 \end{matrix}) = (\begin{matrix} 1 & 3 \\ 2 & - 1 \end{matrix})

Penyelesaian:

\begin{array}{l} (\begin{matrix} 1 & x + 2 \\ 4 - y & - 1 \end{matrix}) = (\begin{matrix} 1 & 3 \\ 2 & - 1 \end{matrix}) \\ x + 2 = 3 \\ x = 1 \\ 4 - y = 2 \\ - y = - 2 \\ y = 2 \end{array}

Contoh 2:

Hitung nilai p dan nilai q dalam setiap persamaan matriks yang berikut:

\begin{array}{l} (a) (\begin{matrix} 3 & 2 p + q \\ p & - 3 \end{matrix}) = (\begin{matrix} 3 & 1 \\ 8 - 2 q & - 3 \end{matrix}) \\ (b) (\begin{matrix} 10 & 0 \\ 5 p - 8 & 1 \end{matrix}) = (\begin{matrix} p - 2 q & 0 \\ - 4 q & 1 \end{matrix}) \end{array}

Penyelesaian:

(a) (\begin{matrix} 3 & 2 p + q \\ p & - 3 \end{matrix}) = (\begin{matrix} 3 & 1 \\ 8 - 2 q & - 3 \end{matrix})

2p + q = 1

q= 1 – 2p ----(1)

p= 8 – 2q ----(2)

Gantikan (1) ke dalam (2),

p= 8 – 2 (1 – 2p)

p= 8 – 2 + 4p

p– 4p = 6

–3p = 6

p= –2

Gantikan p= –2 ke dalam (1),

q= 1 – 2(–2)

q= 5

(b) (\begin{matrix} 10 & 0 \\ 5 p - 8 & 1 \end{matrix}) = (\begin{matrix} p - 2 q & 0 \\ - 4 q & 1 \end{matrix})

10 = p – 2 q

p= 10 + 2q ----(1)

5p – 8 = –4q ----(2)

Gantikan (1) ke dalam (2),

5 (10 + 2q) – 8 = –4q

50 + 10q – 8 = –4q

14q = –42

q= –3

Gantikan q= –3 ke dalam (1),

p= 10 + 2(–3)

p= 4

Bab 15 Matriks

Posted on April 30, 2016 by user

4.10 SPM Practis (Soalan Panjang)

Soalan 1:

Diberi bahawa matriks A =

(\begin{matrix} 3 & - 1 \\ 5 & - 2 \end{matrix})

(a) Cari matriks songsang bagi A.

(b) Tulis persamaan linear serentak berikut dalam persamaan matriks:

3u – v = 9

5u – 2v = 13

Seterusnya, menggunakan kaedah matriks, hitung nilai u dan nilai v.

Penyelesaian:

\begin{array}{l} (a) \\ A^{- 1} = \frac{1}{3 (- 2) - (5) (- 1)} (\begin{matrix} - 2 & 1 \\ - 5 & 3 \end{matrix}) \\ = - 1 (\begin{matrix} - 2 & 1 \\ - 5 & 3 \end{matrix}) = (\begin{matrix} 2 & - 1 \\ 5 & - 3 \end{matrix}) \end{array}

\begin{array}{l} (b) \\ (\begin{matrix} 3 & - 1 \\ 5 & - 2 \end{matrix}) (\begin{array}{l} u \\ v \end{array}) = (\begin{array}{l} 9 \\ 13 \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = - 1 (\begin{matrix} - 2 & 1 \\ - 5 & 3 \end{matrix}) (\begin{array}{l} 9 \\ 13 \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = - 1 (\begin{array}{l} (- 2) (9) + (1) (13) \\ (- 5) (9) + (3) (13) \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = - 1 (\begin{array}{l} - 5 \\ - 6 \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = (\begin{array}{l} 5 \\ 6 \end{array}) \\ ∴ u = 5, v = 6 \end{array}

Soalan 2:

Diberi bahawa matriks A =

(\begin{matrix} 2 & - 5 \\ 1 & 3 \end{matrix})

dan matriks B =

m (\begin{matrix} 3 & k \\ - 1 & 2 \end{matrix})

dengan keadaan AB =

(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

(a) Cari nilai m dan nilai k.

(b) Tulis persamaan linear serentak berikut dalam persamaan matriks:

2u – 5v = –15

u+ 3v = –2

Seterusnya, menggunakan kaedah matriks, hitung nilai u dan nilai v.

Penyelesaian:

(a)

AB=

(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

, Songsang bagi matriks A ialah B.

m = \frac{1}{(2) (3) - (- 5) (1)} = \frac{1}{11}

k= 5

(b)

\begin{array}{l} (\begin{matrix} 2 & - 5 \\ 1 & 3 \end{matrix}) (\begin{array}{l} u \\ v \end{array}) = (\begin{array}{l} - 15 \\ - 2 \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = \frac{1}{11} (\begin{matrix} 3 & 5 \\ - 1 & 2 \end{matrix}) (\begin{array}{l} - 15 \\ - 2 \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = \frac{1}{11} (\begin{array}{l} (3) (- 15) + (5) (- 2) \\ (- 1) (- 15) + (2) (- 2) \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = \frac{1}{11} (\begin{array}{l} - 55 \\ 11 \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = (\begin{array}{l} - 5 \\ 1 \end{array}) \\ ∴ u = - 5, v = 1 \end{array}

Bab 15 Matriks

Posted on April 30, 2016 by user

Soalan 3:

Diberi bahawa

Q (\begin{matrix} 3 & 2 \\ 6 & 5 \end{matrix}) = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

, dengan keadaan Q ialah matriks 2 × 2.

(a) Cari matriks Q.

(b) Tulis persamaan linear serentak berikut dalam persamaan matriks:

3u + 2v = 5

6u + 5v = 2

Seterusnya, menggunakan kaedah matriks, hitung nilai u dan nilai v.

Penyelesaian:

\begin{array}{l} (a) \\ Q = {(\begin{matrix} 3 & 2 \\ 6 & 5 \end{matrix})}^{- 1} \\ Q = \frac{1}{3 (5) - 2 (6)} (\begin{matrix} 5 & - 2 \\ - 6 & 3 \end{matrix}) \\ Q = \frac{1}{3} (\begin{matrix} 5 & - 2 \\ - 6 & 3 \end{matrix}) \\ Q = (\begin{matrix} \frac{5}{3} & - \frac{2}{3} \\ - 2 & 1 \end{matrix}) \end{array}

\begin{array}{l} (b) \\ (\begin{matrix} 3 & 2 \\ 6 & 5 \end{matrix}) (\begin{array}{l} u \\ v \end{array}) = (\begin{array}{l} 5 \\ 2 \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = \frac{1}{3} (\begin{matrix} 5 & - 2 \\ - 6 & 3 \end{matrix}) (\begin{array}{l} 5 \\ 2 \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = \frac{1}{3} (\begin{array}{l} (5) (5) + (- 2) (2) \\ (- 6) (5) + (3) (2) \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = \frac{1}{3} (\begin{array}{l} 21 \\ - 24 \end{array}) \\ (\begin{array}{l} u \\ v \end{array}) = (\begin{array}{l} 7 \\ - 8 \end{array}) \\ ∴ u = 7, v = - 8 \end{array}

Soalan 4:

Diberi bahawa

Q (\begin{matrix} 3 & - 2 \\ 5 & - 4 \end{matrix}) = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

, dengan keadaan Q ialah matriks 2 × 2.

(a) Cari matriks Q.

(b) Tulis persamaan linear serentak berikut dalam persamaan matriks:

3x – 2y = 7

5x – 4y = 9

Seterusnya, menggunakan kaedah matriks, hitung nilai x dan nilai y.

Penyelesaian:

\begin{array}{l} (a) \\ Q = \frac{1}{3 (- 4) - (5) (- 2)} (\begin{matrix} - 4 & 2 \\ - 5 & 3 \end{matrix}) \\ = - \frac{1}{2} (\begin{matrix} - 4 & 2 \\ - 5 & 3 \end{matrix}) \\ = (\begin{matrix} 2 & - 1 \\ \frac{5}{2} & - \frac{3}{2} \end{matrix}) \end{array}

\begin{array}{l} (b) \\ (\begin{matrix} 3 & - 2 \\ 5 & - 4 \end{matrix}) (\begin{array}{l} x \\ y \end{array}) = (\begin{array}{l} 7 \\ 9 \end{array}) \\ (\begin{array}{l} x \\ y \end{array}) = - \frac{1}{2} (\begin{matrix} - 4 & 2 \\ - 5 & 3 \end{matrix}) (\begin{array}{l} 7 \\ 9 \end{array}) \\ (\begin{array}{l} x \\ y \end{array}) = - \frac{1}{2} (\begin{array}{l} (- 4) (7) + (2) (9) \\ (- 5) (7) + (3) (9) \end{array}) \\ (\begin{array}{l} x \\ y \end{array}) = - \frac{1}{2} (\begin{array}{l} - 10 \\ - 8 \end{array}) \\ (\begin{array}{l} x \\ y \end{array}) = (\begin{array}{l} 5 \\ 4 \end{array}) \\ ∴ x = 5, y = 4 \end{array}

Bab 15 Matriks

Posted on April 30, 2016 by user

4.9 SPM Practis (Soalan Pendek)

Soalan 1:

(\begin{matrix} 1 & 4 \\ 6 & 2 \end{matrix}) + 3 (\begin{matrix} 2 & 0 \\ 4 & - 3 \end{matrix}) - (\begin{matrix} - 3 & 0 \\ - 2 & - 5 \end{matrix}) =

Penyelesaian:

\begin{array}{l} (\begin{matrix} 1 & 4 \\ 6 & 2 \end{matrix}) + 3 (\begin{matrix} 2 & 0 \\ 4 & - 3 \end{matrix}) - (\begin{matrix} - 3 & 0 \\ - 2 & - 5 \end{matrix}) \\ = (\begin{matrix} 1 & 4 \\ 6 & 2 \end{matrix}) + (\begin{matrix} 6 & 0 \\ 12 & - 9 \end{matrix}) - (\begin{matrix} - 3 & 0 \\ - 2 & - 5 \end{matrix}) \\ = (\begin{matrix} 7 & 4 \\ 18 & - 7 \end{matrix}) - (\begin{matrix} - 3 & 0 \\ - 2 & - 5 \end{matrix}) \\ = (\begin{matrix} 7 - (- 3) & 4 - 0 \\ 18 - (- 2) & - 7 - (- 5) \end{matrix}) \\ = (\begin{matrix} 10 & 4 \\ 20 & - 2 \end{matrix}) \end{array}

Soalan 2:

Cari nilai mdalam persamaan matriks berikut:

(\begin{matrix} 9 & - 4 \\ 5 & 0 \end{matrix}) + \frac{1}{2} (\begin{matrix} 8 & m \\ 6 & - 10 \end{matrix}) = (\begin{matrix} 13 & - 7 \\ - 2 & 1 \end{matrix})

Penyelesaian:

\begin{array}{l} (\begin{matrix} 9 & - 4 \\ 5 & 0 \end{matrix}) + \frac{1}{2} (\begin{matrix} 8 & m \\ 6 & - 10 \end{matrix}) = (\begin{matrix} 13 & - 7 \\ - 2 & 1 \end{matrix}) \\ - 4 + \frac{1}{2} m = - 7 \\ \frac{1}{2} m = - 3 \\ m = - 6 \end{array}

Soalan 3:

\begin{array}{l} Diberi \\ (\begin{array}{l} 2 x \\ 3 y \end{array}) - 4 (\begin{array}{l} - 2 \\ 3 \end{array}) = (\begin{array}{l} - 2 \\ 6 \end{array}) \\ Cari nilai x dan y . \end{array}

Penyelesaian:

2x + 8 = –2

2x = –10

x= –5

3y – 12 = 6

3y = 18

y= 6

Soalan 4:

Diberi bahawa persamaan matriks 3(6 m) + n(3 4) = (12 7),

Cari nilai m+ n.

Penyelesaian:

3(6 m) + n(3 4) = (12 7)

18 + 3n = 12

3n = –6

n= –2

3m + 4n = 7

3m + 4(–2) = 7

3m = 15

m= 5

Maka m + n = 5 + (–2) = 3