Soalan 9:
$$ \text { Diberi matriks } F=\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right] $$
(a) Hitung F3.
(b) Seterusnya, hitung matriks G jika F3 – 5G = 12F
Penyelesaian:
(a)
$$ \begin{aligned} F^3 & =F F F \\ & =\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right]\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right]\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} -1(-1)+2(3) & -1(2)+2(-4) \\ 3(-1)+(-4)(3) & 3(2)+(-4)(-4) \end{array}\right]\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} 7 & -10 \\ -15 & 22 \end{array}\right]\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} 7(-1)+(-10)(3) & 7(2)+(-10)(-4) \\ -15(-1)+22(3) & -15(2)+22(-4) \end{array}\right] \\ & =\left[\begin{array}{cc} -37 & 54 \\ 81 & -118 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} F^3-5 G & =12 F \\ {\left[\begin{array}{cc} -37 & 54 \\ 81 & -118 \end{array}\right]-5 G } & =12\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right] \\ {\left[\begin{array}{cc} -37 & 54 \\ 81 & -118 \end{array}\right]-5 G } & =\left[\begin{array}{cc} -12 & 24 \\ 36 & -48 \end{array}\right] \\ 5 G & =\left[\begin{array}{cc} -37 & 54 \\ 81 & -118 \end{array}\right]-\left[\begin{array}{cc} -12 & 24 \\ 36 & -48 \end{array}\right] \\ 5 G & =\left[\begin{array}{cc} -25 & 30 \\ 45 & -70 \end{array}\right] \\ G & =\left[\begin{array}{cc} -5 & 6 \\ 9 & -14 \end{array}\right] \end{aligned} $$
$$ \text { Diberi matriks } F=\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right] $$
(a) Hitung F3.
(b) Seterusnya, hitung matriks G jika F3 – 5G = 12F
Penyelesaian:
(a)
$$ \begin{aligned} F^3 & =F F F \\ & =\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right]\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right]\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} -1(-1)+2(3) & -1(2)+2(-4) \\ 3(-1)+(-4)(3) & 3(2)+(-4)(-4) \end{array}\right]\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} 7 & -10 \\ -15 & 22 \end{array}\right]\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} 7(-1)+(-10)(3) & 7(2)+(-10)(-4) \\ -15(-1)+22(3) & -15(2)+22(-4) \end{array}\right] \\ & =\left[\begin{array}{cc} -37 & 54 \\ 81 & -118 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} F^3-5 G & =12 F \\ {\left[\begin{array}{cc} -37 & 54 \\ 81 & -118 \end{array}\right]-5 G } & =12\left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \end{array}\right] \\ {\left[\begin{array}{cc} -37 & 54 \\ 81 & -118 \end{array}\right]-5 G } & =\left[\begin{array}{cc} -12 & 24 \\ 36 & -48 \end{array}\right] \\ 5 G & =\left[\begin{array}{cc} -37 & 54 \\ 81 & -118 \end{array}\right]-\left[\begin{array}{cc} -12 & 24 \\ 36 & -48 \end{array}\right] \\ 5 G & =\left[\begin{array}{cc} -25 & 30 \\ 45 & -70 \end{array}\right] \\ G & =\left[\begin{array}{cc} -5 & 6 \\ 9 & -14 \end{array}\right] \end{aligned} $$
Soalan 10:
$$ \text { Diberi } \frac{1}{p}\left[\begin{array}{ll} 2 & q \\ 1 & 4 \end{array}\right]\left[\begin{array}{cc} 4 & -6 \\ -1 & 2 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$
$$ \text { hitung nilai } p \text { dan nilai } q \text {. } $$
Penyelesaian:
$$ \frac{1}{p}\left[\begin{array}{ll} 2 & q \\ 1 & 4 \end{array}\right] \text { ialah matriks songsang bagi }\left[\begin{array}{cc} 4 & -6 \\ -1 & 2 \end{array}\right] \text {. } $$
$$ \begin{aligned} &\text { Maka matriks songsang }\left[\begin{array}{cc} 4 & -6 \\ -1 & 2 \end{array}\right]\\ &\begin{aligned} & =\frac{1}{4(2)-(-6)(-1)}\left[\begin{array}{ll} 2 & 6 \\ 1 & 4 \end{array}\right] \\ & =\frac{1}{2}\left[\begin{array}{ll} 2 & 6 \\ 1 & 4 \end{array}\right] \end{aligned} \end{aligned} $$
$$ \text { Bandingkan } \frac{1}{p}\left[\begin{array}{ll} 2 & q \\ 1 & 4 \end{array}\right] \text { dengan } \frac{1}{2}\left[\begin{array}{ll} 2 & 6 \\ 1 & 4 \end{array}\right] $$
$$ p=2, q=6 . $$
$$ \text { Diberi } \frac{1}{p}\left[\begin{array}{ll} 2 & q \\ 1 & 4 \end{array}\right]\left[\begin{array}{cc} 4 & -6 \\ -1 & 2 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$
$$ \text { hitung nilai } p \text { dan nilai } q \text {. } $$
Penyelesaian:
$$ \frac{1}{p}\left[\begin{array}{ll} 2 & q \\ 1 & 4 \end{array}\right] \text { ialah matriks songsang bagi }\left[\begin{array}{cc} 4 & -6 \\ -1 & 2 \end{array}\right] \text {. } $$
$$ \begin{aligned} &\text { Maka matriks songsang }\left[\begin{array}{cc} 4 & -6 \\ -1 & 2 \end{array}\right]\\ &\begin{aligned} & =\frac{1}{4(2)-(-6)(-1)}\left[\begin{array}{ll} 2 & 6 \\ 1 & 4 \end{array}\right] \\ & =\frac{1}{2}\left[\begin{array}{ll} 2 & 6 \\ 1 & 4 \end{array}\right] \end{aligned} \end{aligned} $$
$$ \text { Bandingkan } \frac{1}{p}\left[\begin{array}{ll} 2 & q \\ 1 & 4 \end{array}\right] \text { dengan } \frac{1}{2}\left[\begin{array}{ll} 2 & 6 \\ 1 & 4 \end{array}\right] $$
$$ p=2, q=6 . $$
Soalan 11:
(a) Tuliskan persamaan linear serentak di bawah dalam bentuk matriks.
(b) Hitung nilai x dan nilai y dengan menggunakan kaedah matriks.
2y – x = 5
3x – 8y = –19
Penyelesaian:
(a)
$$ \begin{aligned} & 2 y-x=5 \\ &-x+2 y=5 \\ & 3 x-8 y=-19 \\ & {\left[\begin{array}{cc} -1 & 2 \\ 3 & -8 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 5 \\ -19 \end{array}\right] } \end{aligned} $$
(b)
$$ \begin{aligned} &\begin{aligned} & {\left[\begin{array}{cc} -1 & 2 \\ 3 & -8 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 5 \\ -19 \end{array}\right]} \\ & {\left[\begin{array}{l} x \\ y \end{array}\right]=\frac{1}{(-1)(-8)-2(3)}\left[\begin{array}{ll} -8 & -2 \\ -3 & -1 \end{array}\right]\left[\begin{array}{c} 5 \\ -19 \end{array}\right]} \\ & =\frac{1}{2}\left[\begin{array}{l} -8(5)+(-2)(-19) \\ -3(5)+(-1)(-19) \end{array}\right] \\ & =\frac{1}{2}\left[\begin{array}{c} -2 \\ 4 \end{array}\right] \\ & =\left[\begin{array}{c} -1 \\ 2 \end{array}\right] \end{aligned}\\ &\text { Maka, } x=-1, y=2 \end{aligned} $$
(a) Tuliskan persamaan linear serentak di bawah dalam bentuk matriks.
(b) Hitung nilai x dan nilai y dengan menggunakan kaedah matriks.
2y – x = 5
3x – 8y = –19
Penyelesaian:
(a)
$$ \begin{aligned} & 2 y-x=5 \\ &-x+2 y=5 \\ & 3 x-8 y=-19 \\ & {\left[\begin{array}{cc} -1 & 2 \\ 3 & -8 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 5 \\ -19 \end{array}\right] } \end{aligned} $$
(b)
$$ \begin{aligned} &\begin{aligned} & {\left[\begin{array}{cc} -1 & 2 \\ 3 & -8 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 5 \\ -19 \end{array}\right]} \\ & {\left[\begin{array}{l} x \\ y \end{array}\right]=\frac{1}{(-1)(-8)-2(3)}\left[\begin{array}{ll} -8 & -2 \\ -3 & -1 \end{array}\right]\left[\begin{array}{c} 5 \\ -19 \end{array}\right]} \\ & =\frac{1}{2}\left[\begin{array}{l} -8(5)+(-2)(-19) \\ -3(5)+(-1)(-19) \end{array}\right] \\ & =\frac{1}{2}\left[\begin{array}{c} -2 \\ 4 \end{array}\right] \\ & =\left[\begin{array}{c} -1 \\ 2 \end{array}\right] \end{aligned}\\ &\text { Maka, } x=-1, y=2 \end{aligned} $$