Soalan 1:
Tentukan sama ada matriks berikut ialah matriks songsang antara satu sama lain.
$$ \text { (a) }\left[\begin{array}{ll} 5 & 4 \\ 3 & 2 \end{array}\right],\left[\begin{array}{cc} 2 & -4 \\ -3 & 5 \end{array}\right] $$
$$ \text { (b) }\left[\begin{array}{ll} 1 & 3 \\ 1 & 4 \end{array}\right],\left[\begin{array}{ll} \frac{1}{2} & 3 \\ 1 & 2 \end{array}\right] $$
$$ \text { (c) }\left[\begin{array}{ll} 1 & 2 \\ 4 & 9 \end{array}\right],\left[\begin{array}{cc} 1 & -2 \\ -4 & 9 \end{array}\right] $$
$$ \text { (d) }\left[\begin{array}{ll} -2 & 3 \\ -5 & 7 \end{array}\right],\left[\begin{array}{ll} 7 & -3 \\ 5 & -2 \end{array}\right] $$
Penyelesaian:
(a)
$$ \begin{aligned} {\left[\begin{array}{ll} 5 & 4 \\ 3 & 2 \end{array}\right]\left[\begin{array}{cc} 2 & -4 \\ -3 & 5 \end{array}\right] } & =\left[\begin{array}{ll} 5(2)+4(-3) & 5(-4)+4(5) \\ 3(2)+2(-3) & 3(-4)+2(5) \end{array}\right] \\ & =\left[\begin{array}{cc} -2 & 0 \\ 0 & -2 \end{array}\right] \end{aligned} $$
$$ \left[\begin{array}{cc} 2 & -4 \\ -3 & 5 \end{array}\right] \text { bukan matriks songsang bagi }\left[\begin{array}{ll} 5 & 4 \\ 3 & 2 \end{array}\right] \text { kerana } $$
$$ \text { hasil darab dua matriks ini bukan matriks identiti. } $$
(b)
$$ \begin{aligned} &\begin{aligned} {\left[\begin{array}{ll} 1 & 3 \\ 1 & 4 \end{array}\right]\left[\begin{array}{ll} \frac{1}{2} & 3 \\ 1 & 2 \end{array}\right] } & =\left[\begin{array}{ll} 1\left(\frac{1}{2}\right)+3(1) & 1(3)+3(2) \\ 1\left(\frac{1}{2}\right)+4(1) & 1(3)+4(2) \end{array}\right] \\ & =\left[\begin{array}{cc} \frac{1}{2} & 9 \\ \frac{9}{2} & 11 \end{array}\right] \end{aligned}\\ &\text { Bukan matriks songsang. } \end{aligned} $$
(c)
$$ \begin{aligned} &\begin{aligned} {\left[\begin{array}{ll} 1 & 2 \\ 4 & 9 \end{array}\right]\left[\begin{array}{cc} 1 & -2 \\ -4 & 9 \end{array}\right] } & =\left[\begin{array}{cc} 1(1)+2(-4) & 1(-2)+2(9) \\ 4(1)+9(-4) & 4(-2)+9(9) \end{array}\right] \\ & =\left[\begin{array}{cc} -7 & 16 \\ -32 & 73 \end{array}\right] \end{aligned}\\ &\text { Bukan matriks songsang. } \end{aligned} $$
(d)
$$ \begin{aligned} {\left[\begin{array}{ll} -2 & 3 \\ -5 & 7 \end{array}\right]\left[\begin{array}{ll} 7 & -3 \\ 5 & -2 \end{array}\right] } & =\left[\begin{array}{ll} -2(7)+3(5) & -2(-3)+3(-2) \\ -5(7)+7(5) & -5(-3)+7(-2) \end{array}\right] \\ & =\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} {\left[\begin{array}{ll} 7 & -3 \\ 5 & -2 \end{array}\right]\left[\begin{array}{ll} -2 & 3 \\ -5 & 7 \end{array}\right] } & =\left[\begin{array}{ll} 7(-2)+(-3)(-5) & 7(3)+(-3)(7) \\ 5(-2)+(-2)(-5) & 5(3)+(-2)(7) \end{array}\right] \\ & =\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned} $$
$$ \text { Ya. }\left[\begin{array}{ll} 7 & -3 \\ 5 & -2 \end{array}\right] \text { ialah matriks songsang bagi }\left[\begin{array}{ll} -2 & 3 \\ -5 & 7 \end{array}\right] \text { kerana } $$
$$ \text { hasil darab dua matriks ini ialah matriks identiti. } $$
Tentukan sama ada matriks berikut ialah matriks songsang antara satu sama lain.
$$ \text { (a) }\left[\begin{array}{ll} 5 & 4 \\ 3 & 2 \end{array}\right],\left[\begin{array}{cc} 2 & -4 \\ -3 & 5 \end{array}\right] $$
$$ \text { (b) }\left[\begin{array}{ll} 1 & 3 \\ 1 & 4 \end{array}\right],\left[\begin{array}{ll} \frac{1}{2} & 3 \\ 1 & 2 \end{array}\right] $$
$$ \text { (c) }\left[\begin{array}{ll} 1 & 2 \\ 4 & 9 \end{array}\right],\left[\begin{array}{cc} 1 & -2 \\ -4 & 9 \end{array}\right] $$
$$ \text { (d) }\left[\begin{array}{ll} -2 & 3 \\ -5 & 7 \end{array}\right],\left[\begin{array}{ll} 7 & -3 \\ 5 & -2 \end{array}\right] $$
Penyelesaian:
(a)
$$ \begin{aligned} {\left[\begin{array}{ll} 5 & 4 \\ 3 & 2 \end{array}\right]\left[\begin{array}{cc} 2 & -4 \\ -3 & 5 \end{array}\right] } & =\left[\begin{array}{ll} 5(2)+4(-3) & 5(-4)+4(5) \\ 3(2)+2(-3) & 3(-4)+2(5) \end{array}\right] \\ & =\left[\begin{array}{cc} -2 & 0 \\ 0 & -2 \end{array}\right] \end{aligned} $$
$$ \left[\begin{array}{cc} 2 & -4 \\ -3 & 5 \end{array}\right] \text { bukan matriks songsang bagi }\left[\begin{array}{ll} 5 & 4 \\ 3 & 2 \end{array}\right] \text { kerana } $$
$$ \text { hasil darab dua matriks ini bukan matriks identiti. } $$
(b)
$$ \begin{aligned} &\begin{aligned} {\left[\begin{array}{ll} 1 & 3 \\ 1 & 4 \end{array}\right]\left[\begin{array}{ll} \frac{1}{2} & 3 \\ 1 & 2 \end{array}\right] } & =\left[\begin{array}{ll} 1\left(\frac{1}{2}\right)+3(1) & 1(3)+3(2) \\ 1\left(\frac{1}{2}\right)+4(1) & 1(3)+4(2) \end{array}\right] \\ & =\left[\begin{array}{cc} \frac{1}{2} & 9 \\ \frac{9}{2} & 11 \end{array}\right] \end{aligned}\\ &\text { Bukan matriks songsang. } \end{aligned} $$
(c)
$$ \begin{aligned} &\begin{aligned} {\left[\begin{array}{ll} 1 & 2 \\ 4 & 9 \end{array}\right]\left[\begin{array}{cc} 1 & -2 \\ -4 & 9 \end{array}\right] } & =\left[\begin{array}{cc} 1(1)+2(-4) & 1(-2)+2(9) \\ 4(1)+9(-4) & 4(-2)+9(9) \end{array}\right] \\ & =\left[\begin{array}{cc} -7 & 16 \\ -32 & 73 \end{array}\right] \end{aligned}\\ &\text { Bukan matriks songsang. } \end{aligned} $$
(d)
$$ \begin{aligned} {\left[\begin{array}{ll} -2 & 3 \\ -5 & 7 \end{array}\right]\left[\begin{array}{ll} 7 & -3 \\ 5 & -2 \end{array}\right] } & =\left[\begin{array}{ll} -2(7)+3(5) & -2(-3)+3(-2) \\ -5(7)+7(5) & -5(-3)+7(-2) \end{array}\right] \\ & =\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} {\left[\begin{array}{ll} 7 & -3 \\ 5 & -2 \end{array}\right]\left[\begin{array}{ll} -2 & 3 \\ -5 & 7 \end{array}\right] } & =\left[\begin{array}{ll} 7(-2)+(-3)(-5) & 7(3)+(-3)(7) \\ 5(-2)+(-2)(-5) & 5(3)+(-2)(7) \end{array}\right] \\ & =\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned} $$
$$ \text { Ya. }\left[\begin{array}{ll} 7 & -3 \\ 5 & -2 \end{array}\right] \text { ialah matriks songsang bagi }\left[\begin{array}{ll} -2 & 3 \\ -5 & 7 \end{array}\right] \text { kerana } $$
$$ \text { hasil darab dua matriks ini ialah matriks identiti. } $$