Soalan 5:
$$ \text { Diberi } G=\left[\begin{array}{cc} p & 5 \\ 1 & -4 \end{array}\right] \text { dan } H=\left[\begin{array}{cc} -6 & 7 \\ 3 & 0 \end{array}\right] \text {, hitung nilai } p \text {, nilai } q \text { dan nilai } r \text { jika } $$
$$ \text { (a) } \quad G H=\left[\begin{array}{cc} 3 & 2 q \\ -18 & 3 p+r \end{array}\right] $$
$$ \text { (b) } \quad G^2=\left[\begin{array}{cc} r & -25 \\ -5 & 7 q \end{array}\right] $$
$$ \text { (c) } \quad H G=\left[\begin{array}{cc} -11 & 2.5 q \\ \frac{p+3 r}{2} & 5 p \end{array}\right] $$
$$ \text { (d) } H^2=\left[\begin{array}{cc} 57 & 6 p \\ 1.2 q & \frac{7 r}{5} \end{array}\right] $$
Penyelesaian:
(a)
$$ \begin{aligned} G H & =\left[\begin{array}{cc} 3 & 2 q \\ -18 & 3 p+r \end{array}\right] \\ {\left[\begin{array}{cc} P & 5 \\ 1 & -4 \end{array}\right]\left[\begin{array}{cc} -6 & 7 \\ 3 & 0 \end{array}\right] } & =\left[\begin{array}{cc} 3 & 2 q \\ -18 & 3 p+r \end{array}\right] \\ {\left[\begin{array}{cc} P(-6)+5(3) & P(7)+5(0) \\ 1(-6)+(-4) 3 & 1(7)+(-4) 0 \end{array}\right] } & =\left[\begin{array}{cc} 3 & 2 q \\ -18 & 3 p+r \end{array}\right] \\ {\left[\begin{array}{cc} -6 p+15 & 7 p \\ -18 & 7 \end{array}\right] } & =\left[\begin{array}{cc} 3 & 2 q \\ -18 & 3 p+r \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} -6 p+15 & =3 \\ -6 p & =-12 \\ p & =2 \end{aligned} \end{aligned} $$
$$ \begin{aligned} 7 p & =2 q \\ 7(2) & =2 q \\ 14 & =2 q \\ q & =7 \end{aligned} $$
$$ \begin{array}{r} 3 p+r=7 \\ 3(2)+r=7 \\ 6+r=7 \\ r=1 \end{array} $$
(b)
$$ \begin{aligned} G^2 & =\left[\begin{array}{cc} r & -25 \\ -5 & 7 q \end{array}\right] \\ {\left[\begin{array}{cc} P & 5 \\ 1 & -4 \end{array}\right]\left[\begin{array}{cc} P & 5 \\ 1 & -4 \end{array}\right] } & =\left[\begin{array}{cc} r & -25 \\ -5 & 7 q \end{array}\right] \\ {\left[\begin{array}{cc} P(P)+5(1) & P(5)+5(-4) \\ 1(p)+(-4) 1 & 1(5)+(-4)(-4) \end{array}\right] } & =\left[\begin{array}{cc} r & -25 \\ -5 & 7 q \end{array}\right] \\ {\left[\begin{array}{cc} P^2+5 & 5 p-20 \\ p-4 & 21 \end{array}\right] } & =\left[\begin{array}{cc} r & -25 \\ -5 & 7 q \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} p-4 & =-5 \\ p & =-1 \end{aligned}\\ &\begin{aligned} 7 q & =21 \\ q & =3 \end{aligned}\\ &\begin{aligned} p^2+5 & =r \\ (-1)^2+5 & =r \\ r & =6 \end{aligned} \end{aligned} $$
(c)
$$ \begin{aligned} H G & =\left[\begin{array}{cc} -11 & 2.5 q \\ \frac{p+3 r}{2} & 5 p \end{array}\right] \\ {\left[\begin{array}{cc} -6 & 7 \\ 3 & 0 \end{array}\right]\left[\begin{array}{cc} p & 5 \\ 1 & -4 \end{array}\right] } & =\left[\begin{array}{cc} -11 & 2.5 q \\ \frac{p+3 r}{2} & 5 p \end{array}\right] \\ {\left[\begin{array}{cc} -6(p)+7(1) & (-6) 5+7(-4) \\ 3(p)+0(1) & 3(5)+0(-4) \end{array}\right] } & =\left[\begin{array}{cc} -11 & 2.5 q \\ \frac{p+3 r}{2} & 5 p \end{array}\right] \\ {\left[\begin{array}{cc} -6 p+7 & -58 \\ 3 p & 15 \end{array}\right] } & =\left[\begin{array}{cc} -11 & 2.5 q \\ \frac{p+3 r}{2} & 5 p \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} 5 p & =15 \\ p & =3 \end{aligned}\\ &\begin{aligned} 2.5 q & =-58 \\ q & =-23.2 \end{aligned} \end{aligned} $$
$$ \begin{aligned} 3 p & =\frac{p+3 r}{2} \\ 3(3) & =\frac{3+3 r}{2} \\ 18 & =3+3 r \\ 15 & =3 r \\ r & =5 \end{aligned} $$
(d)
$$ \begin{aligned} H^2 & =\left[\begin{array}{cc} 57 & 6 p \\ 1.2 q & \frac{7 r}{5} \end{array}\right] \\ {\left[\begin{array}{cc} -6 & 7 \\ 3 & 0 \end{array}\right]\left[\begin{array}{cc} -6 & 7 \\ 3 & 0 \end{array}\right] } & =\left[\begin{array}{cc} 57 & 6 p \\ 1.2 q & \frac{7 r}{5} \end{array}\right] \\ {\left[\begin{array}{cc} (-6)(-6)+7(3) & (-6) 7+7(0) \\ 3(-6)+0(3) & 3(7)+0(0) \end{array}\right] } & =\left[\begin{array}{cc} 57 & 6 p \\ 1.29 & \frac{7 r}{5} \end{array}\right] \\ {\left[\begin{array}{cc} 57 & -42 \\ -18 & 21 \end{array}\right] } & =\left[\begin{array}{cc} 57 & 6 p \\ 1.2 q & \frac{7 r}{5} \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} 6 p & =-42 \\ p & =-7 \\ 1.2 q & =-18 \\ q & =-15 \end{aligned} \end{aligned} $$
$$ \begin{aligned} \frac{7 r}{5} & =21 \\ 7 r & =105 \\ r & =15 \end{aligned} $$
$$ \text { Diberi } G=\left[\begin{array}{cc} p & 5 \\ 1 & -4 \end{array}\right] \text { dan } H=\left[\begin{array}{cc} -6 & 7 \\ 3 & 0 \end{array}\right] \text {, hitung nilai } p \text {, nilai } q \text { dan nilai } r \text { jika } $$
$$ \text { (a) } \quad G H=\left[\begin{array}{cc} 3 & 2 q \\ -18 & 3 p+r \end{array}\right] $$
$$ \text { (b) } \quad G^2=\left[\begin{array}{cc} r & -25 \\ -5 & 7 q \end{array}\right] $$
$$ \text { (c) } \quad H G=\left[\begin{array}{cc} -11 & 2.5 q \\ \frac{p+3 r}{2} & 5 p \end{array}\right] $$
$$ \text { (d) } H^2=\left[\begin{array}{cc} 57 & 6 p \\ 1.2 q & \frac{7 r}{5} \end{array}\right] $$
Penyelesaian:
(a)
$$ \begin{aligned} G H & =\left[\begin{array}{cc} 3 & 2 q \\ -18 & 3 p+r \end{array}\right] \\ {\left[\begin{array}{cc} P & 5 \\ 1 & -4 \end{array}\right]\left[\begin{array}{cc} -6 & 7 \\ 3 & 0 \end{array}\right] } & =\left[\begin{array}{cc} 3 & 2 q \\ -18 & 3 p+r \end{array}\right] \\ {\left[\begin{array}{cc} P(-6)+5(3) & P(7)+5(0) \\ 1(-6)+(-4) 3 & 1(7)+(-4) 0 \end{array}\right] } & =\left[\begin{array}{cc} 3 & 2 q \\ -18 & 3 p+r \end{array}\right] \\ {\left[\begin{array}{cc} -6 p+15 & 7 p \\ -18 & 7 \end{array}\right] } & =\left[\begin{array}{cc} 3 & 2 q \\ -18 & 3 p+r \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} -6 p+15 & =3 \\ -6 p & =-12 \\ p & =2 \end{aligned} \end{aligned} $$
$$ \begin{aligned} 7 p & =2 q \\ 7(2) & =2 q \\ 14 & =2 q \\ q & =7 \end{aligned} $$
$$ \begin{array}{r} 3 p+r=7 \\ 3(2)+r=7 \\ 6+r=7 \\ r=1 \end{array} $$
(b)
$$ \begin{aligned} G^2 & =\left[\begin{array}{cc} r & -25 \\ -5 & 7 q \end{array}\right] \\ {\left[\begin{array}{cc} P & 5 \\ 1 & -4 \end{array}\right]\left[\begin{array}{cc} P & 5 \\ 1 & -4 \end{array}\right] } & =\left[\begin{array}{cc} r & -25 \\ -5 & 7 q \end{array}\right] \\ {\left[\begin{array}{cc} P(P)+5(1) & P(5)+5(-4) \\ 1(p)+(-4) 1 & 1(5)+(-4)(-4) \end{array}\right] } & =\left[\begin{array}{cc} r & -25 \\ -5 & 7 q \end{array}\right] \\ {\left[\begin{array}{cc} P^2+5 & 5 p-20 \\ p-4 & 21 \end{array}\right] } & =\left[\begin{array}{cc} r & -25 \\ -5 & 7 q \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} p-4 & =-5 \\ p & =-1 \end{aligned}\\ &\begin{aligned} 7 q & =21 \\ q & =3 \end{aligned}\\ &\begin{aligned} p^2+5 & =r \\ (-1)^2+5 & =r \\ r & =6 \end{aligned} \end{aligned} $$
(c)
$$ \begin{aligned} H G & =\left[\begin{array}{cc} -11 & 2.5 q \\ \frac{p+3 r}{2} & 5 p \end{array}\right] \\ {\left[\begin{array}{cc} -6 & 7 \\ 3 & 0 \end{array}\right]\left[\begin{array}{cc} p & 5 \\ 1 & -4 \end{array}\right] } & =\left[\begin{array}{cc} -11 & 2.5 q \\ \frac{p+3 r}{2} & 5 p \end{array}\right] \\ {\left[\begin{array}{cc} -6(p)+7(1) & (-6) 5+7(-4) \\ 3(p)+0(1) & 3(5)+0(-4) \end{array}\right] } & =\left[\begin{array}{cc} -11 & 2.5 q \\ \frac{p+3 r}{2} & 5 p \end{array}\right] \\ {\left[\begin{array}{cc} -6 p+7 & -58 \\ 3 p & 15 \end{array}\right] } & =\left[\begin{array}{cc} -11 & 2.5 q \\ \frac{p+3 r}{2} & 5 p \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} 5 p & =15 \\ p & =3 \end{aligned}\\ &\begin{aligned} 2.5 q & =-58 \\ q & =-23.2 \end{aligned} \end{aligned} $$
$$ \begin{aligned} 3 p & =\frac{p+3 r}{2} \\ 3(3) & =\frac{3+3 r}{2} \\ 18 & =3+3 r \\ 15 & =3 r \\ r & =5 \end{aligned} $$
(d)
$$ \begin{aligned} H^2 & =\left[\begin{array}{cc} 57 & 6 p \\ 1.2 q & \frac{7 r}{5} \end{array}\right] \\ {\left[\begin{array}{cc} -6 & 7 \\ 3 & 0 \end{array}\right]\left[\begin{array}{cc} -6 & 7 \\ 3 & 0 \end{array}\right] } & =\left[\begin{array}{cc} 57 & 6 p \\ 1.2 q & \frac{7 r}{5} \end{array}\right] \\ {\left[\begin{array}{cc} (-6)(-6)+7(3) & (-6) 7+7(0) \\ 3(-6)+0(3) & 3(7)+0(0) \end{array}\right] } & =\left[\begin{array}{cc} 57 & 6 p \\ 1.29 & \frac{7 r}{5} \end{array}\right] \\ {\left[\begin{array}{cc} 57 & -42 \\ -18 & 21 \end{array}\right] } & =\left[\begin{array}{cc} 57 & 6 p \\ 1.2 q & \frac{7 r}{5} \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} 6 p & =-42 \\ p & =-7 \\ 1.2 q & =-18 \\ q & =-15 \end{aligned} \end{aligned} $$
$$ \begin{aligned} \frac{7 r}{5} & =21 \\ 7 r & =105 \\ r & =15 \end{aligned} $$
Soalan 6:
Encik Koh menyewa sebuah gerai di Expo Pendidikan untuk menjual tiga jenis barangan yang ditunjukkan dalam jadual di bawah.
Diberi keuntungan jualan setiap barangan A, B dan C masing-masing ialah RM5, RM8 dan RM6. Hitung jumlah keuntungan yang diterima oleh Encik Koh setiap hari.
Tunjukkan pengiraan anda dalam bentuk matriks.
[Diberi bahawa jumlah keuntungan = jualan barangan A × keuntungan barangan A
+ jualan barangan B × keuntungan barangan B
+ jualan barangan C × keuntungan barangan C]
Penyelesaian:
$$ \begin{aligned} \text { Jumlah keuntungan } & =\left[\begin{array}{lll} 40 & 28 & 36 \\ 42 & 36 & 30 \\ 35 & 25 & 42 \end{array}\right]\left[\begin{array}{l} 5 \\ 8 \\ 6 \end{array}\right] \\ & =\left[\begin{array}{l} 40(5)+28(8)+36(6) \\ 42(5)+36(8)+30(6) \\ 35(5)+25(8)+42(6) \end{array}\right] \\ & =\left[\begin{array}{l} 640 \\ 678 \\ 627 \end{array}\right] \end{aligned} $$
Encik Koh menyewa sebuah gerai di Expo Pendidikan untuk menjual tiga jenis barangan yang ditunjukkan dalam jadual di bawah.
Diberi keuntungan jualan setiap barangan A, B dan C masing-masing ialah RM5, RM8 dan RM6. Hitung jumlah keuntungan yang diterima oleh Encik Koh setiap hari.
Tunjukkan pengiraan anda dalam bentuk matriks.
[Diberi bahawa jumlah keuntungan = jualan barangan A × keuntungan barangan A
+ jualan barangan B × keuntungan barangan B
+ jualan barangan C × keuntungan barangan C]
Penyelesaian:
$$ \begin{aligned} \text { Jumlah keuntungan } & =\left[\begin{array}{lll} 40 & 28 & 36 \\ 42 & 36 & 30 \\ 35 & 25 & 42 \end{array}\right]\left[\begin{array}{l} 5 \\ 8 \\ 6 \end{array}\right] \\ & =\left[\begin{array}{l} 40(5)+28(8)+36(6) \\ 42(5)+36(8)+30(6) \\ 35(5)+25(8)+42(6) \end{array}\right] \\ & =\left[\begin{array}{l} 640 \\ 678 \\ 627 \end{array}\right] \end{aligned} $$