4.10.3 Matriks, SPM Praktis (Soalan Panjang)

Soalan 5:
$\text{(a) Diberi }\frac{1}{14}\left(\begin{array}{cc}2& s\\ -4& t\end{array}\right)\left(\begin{array}{cc}t& -1\\ 4& 2\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\text{ cari nilai }s\text{ dan nilai }t\text{.}$
(b) Tulis persamaan linear serentak berikut dalam bentuk matriks:
3x – 2y = 5
9x + y = 1
Seterusnya, menggunakan kaedah matriks, hitung nilai x dan nilai y.

Penyelesaian:
$\begin{array}{l}\text{(a)}\\ \frac{1}{14}\left(\begin{array}{cc}2& s\\ -4& t\end{array}\right)\left(\begin{array}{cc}t& -1\\ 4& 2\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\\ \frac{1}{14}\left(\begin{array}{cc}2t+4s& -2+2s\\ -4t+4t& 4+2t\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\\ \frac{-2+2s}{14}=0\\ 2s=2\\ s=1\\ \\ \frac{4+2t}{14}=1\\ 4+2t=14\\ 2t=10\\ t=5\end{array}$

$\begin{array}{l}\text{(b)}\\ \left(\begin{array}{cc}3& -2\\ 9& 1\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}5\\ 1\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{21}\left(\begin{array}{cc}1& 2\\ -9& 3\end{array}\right)\left(\begin{array}{l}5\\ 1\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{21}\left(\begin{array}{l}\left(1\right)\left(5\right)+\left(2\right)\left(1\right)\\ \left(-9\right)\left(5\right)+\left(3\right)\left(1\right)\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{21}\left(\begin{array}{l}7\\ -42\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}\frac{1}{3}\\ -2\end{array}\right)\\ \therefore x=\frac{1}{3},y=-2\end{array}$

Soalan 6:
$\begin{array}{l}\text{Diberi bahawa matriks }P=\left(\begin{array}{cc}6& -3\\ -5& 2\end{array}\right)\text{ dan matriks }Q=\frac{1}{m}\left(\begin{array}{cc}2& 3\\ 5& n\end{array}\right)\\ \text{dengan keadaan }PQ=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\text{.}\end{array}$
(a) Cari nilai m dan nilai n.
(b) Tulis persamaan linear serentak berikut dalam bentuk matriks:
6x – 3y = –24
–5x + 2y = 18
Seterusnya, menggunakan kaedah matriks, hitung nilai x dan nilai y.

Penyelesaian:
$\begin{array}{l}\text{(a)}\\ m=6\left(2\right)-\left(-3\right)\left(-5\right)\\ =12-15\\ m=-3\\ n=6\end{array}$

$\begin{array}{l}\text{(b)}\\ \left(\begin{array}{cc}6& -3\\ -5& 2\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}-24\\ 18\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{12-15}\left(\begin{array}{cc}2& 3\\ 5& 6\end{array}\right)\left(\begin{array}{l}-24\\ 18\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{-3}\left(\begin{array}{l}\left(2\right)\left(-24\right)+\left(3\right)\left(18\right)\\ \left(5\right)\left(-24\right)+\left(6\right)\left(18\right)\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{-3}\left(\begin{array}{l}6\\ -12\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}-2\\ 4\end{array}\right)\\ \therefore x=-2,y=4\end{array}$