4.10.5 Matriks, SPM Praktis (Soalan Panjang)

Soalan 9:
$\text{(a) Diberi }\frac{1}{s}\left(\begin{array}{cc}-4& 2\\ -5& 3\end{array}\right)\left(\begin{array}{cc}t& -2\\ 5& -4\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\text{ cari nilai }s\text{ dan nilai }t\text{.}$

(b) Menggunakan kaedah matriks, hitung nilai x dan nilai y yang memuaskan persamaan matriks berikut:
$\left(\begin{array}{cc}-4& 2\\ -5& 3\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}1\\ 2\end{array}\right)$

Penyelesaian:
$\begin{array}{l}\text{(a)}\\ \frac{1}{s}\left(\begin{array}{cc}t& -2\\ 5& -4\end{array}\right)={\left(\begin{array}{cc}-4& 2\\ -5& 3\end{array}\right)}^{-1}\\ =\frac{1}{\left(-4\right)\left(3\right)-\left(2\right)\left(-5\right)}\left(\begin{array}{cc}3& -2\\ 5& -4\end{array}\right)\\ =\frac{1}{-2}\left(\begin{array}{cc}3& -2\\ 5& -4\end{array}\right)\\ \therefore s=-2,t=3\end{array}$

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

Soalan 10 (6 markah):
$\begin{array}{l}\text{Diberi }A=\left(\begin{array}{l}4-2\\ 3-1\end{array}\right),B=m\left(\begin{array}{l}-1n\\ -34\end{array}\right)\\ \text{dan }I=\left(\begin{array}{l}\text{1 0}\\ \text{0 }1\end{array}\right).\end{array}$
(a) Cari nilai m dan nilai n jika AB = I.
(b) Tulis persamaan linear serentak berikut dalam bentuk persamaan matriks:
4x – 2y = 3
3x – y = 2
Seterusnya, menggunakan kaedah matriks, hitung nilai x dan nilai y.

Penyelesaian:
(a)
$\begin{array}{l}\text{Jika }AB=I,\text{ maka }B=A^{-1}\\ A^{-1}=\frac{1}{4\left(-1\right)-\left(-2\right)\left(3\right)}\left(\begin{array}{l}-12\\ -3\text{ 4}\end{array}\right)\\ A^{-1}=\frac{1}{2}\left(\begin{array}{l}-12\\ -3\text{ 4}\end{array}\right)\\ \\ \text{Secara perbandingan:}\\ B=A^{-1}\\ m\left(\begin{array}{l}-1n\\ -34\end{array}\right)=\frac{1}{2}\left(\begin{array}{l}-12\\ -3\text{ 4}\end{array}\right)\\ m=\frac{1}{2};n=2\end{array}$


(b)
$\begin{array}{l}4x-2y=3\\ 3x-y=2\\ \left(\begin{array}{l}4-2\\ 3-1\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}3\\ 2\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{2}\left(\begin{array}{l}-12\\ -3\text{ 4}\end{array}\right)\left(\begin{array}{l}3\\ 2\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{2}\left(\begin{array}{l}\left(-1\right)\left(3\right)\text{+}\left(2\right)\left(2\right)\\ \left(-3\right)\left(3\right)\text{+}\left(4\right)\left(2\right)\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{2}\left(\begin{array}{l}1\\ -1\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}\frac{1}{2}\\ -\frac{1}{2}\end{array}\right)\\ x=\frac{1}{2}\text{ dan }y=-\frac{1}{2}\end{array}$