Soalan 1:
Bagi setiap matriks yang berikut, tentukan sama ada matriks songsang wujud. Jika wujud, hitung matriks songsang.
Penyelesaian:
(a)
$$ \begin{aligned} &\begin{aligned} & {\left[\begin{array}{ll} 6 & 0 \\ 0 & 1 \end{array}\right]} \\ & \begin{aligned} a d-b c & =6(1)-0(0) \\ & =6 \\ & \neq 0 \end{aligned} \end{aligned}\\ &\text { Maka matriks songsang wujud } \end{aligned} $$
$$ \begin{aligned} \text { Matriks songsang } & =\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right] \\ & =\frac{1}{6(1)-0(0)}\left[\begin{array}{ll} 1 & 0 \\ 0 & 6 \end{array}\right] \\ & =\frac{1}{6}\left[\begin{array}{ll} 1 & 0 \\ 0 & 6 \end{array}\right] \\ & =\left[\begin{array}{ll} \frac{1}{6} & 0 \\ 0 & 1 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} &\begin{aligned} & {\left[\begin{array}{ll} 2 & 3 \\ 1 & 2 \end{array}\right]} \\ & \begin{aligned} a d-b c & =2(2)-3(1) \\ & =1 \\ & \neq 0 \end{aligned} \end{aligned}\\ &\text { Maka matriks songsang wujud } \end{aligned} $$
$$ \begin{aligned} \text { Matriks songsang } & =\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right] \\ & =\frac{1}{2(2)-3(1)}\left[\begin{array}{cc} 2 & -3 \\ -1 & 2 \end{array}\right] \\ & =1\left[\begin{array}{cc} 2 & -3 \\ -1 & 2 \end{array}\right] \\ & =\left[\begin{array}{cc} 2 & -3 \\ -1 & 2 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} &\begin{aligned} & {\left[\begin{array}{cc} -2 & 5 \\ 3 & -9 \end{array}\right]} \\ & \begin{aligned} a d-b c & =(-2)(-9)-5(3) \\ & =3 \\ & \neq 0 \end{aligned} \end{aligned}\\ &\text { Maka matriks songsang wujud } \end{aligned} $$
$$ \begin{aligned} \text { Matriks songsang } & =\frac{1}{(-2)(-9)-5(3)}\left[\begin{array}{ll} -9 & -5 \\ -3 & -2 \end{array}\right] \\ & =\frac{1}{3}\left[\begin{array}{ll} -9 & -5 \\ -3 & -2 \end{array}\right] \\ & =\left[\begin{array}{rr} -3 & -\frac{5}{3} \\ -1 & -\frac{2}{3} \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} &\begin{aligned} & {\left[\begin{array}{cc} 4 & 2 \\ 2 & 1 \end{array}\right] } \\ & a d-b c=(4)(1)-2(2) \\ &=0 \end{aligned}\\ &\text { Maka matriks songsang tidak wujud } \end{aligned} $$
Bagi setiap matriks yang berikut, tentukan sama ada matriks songsang wujud. Jika wujud, hitung matriks songsang.
Penyelesaian:
(a)
$$ \begin{aligned} &\begin{aligned} & {\left[\begin{array}{ll} 6 & 0 \\ 0 & 1 \end{array}\right]} \\ & \begin{aligned} a d-b c & =6(1)-0(0) \\ & =6 \\ & \neq 0 \end{aligned} \end{aligned}\\ &\text { Maka matriks songsang wujud } \end{aligned} $$
$$ \begin{aligned} \text { Matriks songsang } & =\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right] \\ & =\frac{1}{6(1)-0(0)}\left[\begin{array}{ll} 1 & 0 \\ 0 & 6 \end{array}\right] \\ & =\frac{1}{6}\left[\begin{array}{ll} 1 & 0 \\ 0 & 6 \end{array}\right] \\ & =\left[\begin{array}{ll} \frac{1}{6} & 0 \\ 0 & 1 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} &\begin{aligned} & {\left[\begin{array}{ll} 2 & 3 \\ 1 & 2 \end{array}\right]} \\ & \begin{aligned} a d-b c & =2(2)-3(1) \\ & =1 \\ & \neq 0 \end{aligned} \end{aligned}\\ &\text { Maka matriks songsang wujud } \end{aligned} $$
$$ \begin{aligned} \text { Matriks songsang } & =\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right] \\ & =\frac{1}{2(2)-3(1)}\left[\begin{array}{cc} 2 & -3 \\ -1 & 2 \end{array}\right] \\ & =1\left[\begin{array}{cc} 2 & -3 \\ -1 & 2 \end{array}\right] \\ & =\left[\begin{array}{cc} 2 & -3 \\ -1 & 2 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} &\begin{aligned} & {\left[\begin{array}{cc} -2 & 5 \\ 3 & -9 \end{array}\right]} \\ & \begin{aligned} a d-b c & =(-2)(-9)-5(3) \\ & =3 \\ & \neq 0 \end{aligned} \end{aligned}\\ &\text { Maka matriks songsang wujud } \end{aligned} $$
$$ \begin{aligned} \text { Matriks songsang } & =\frac{1}{(-2)(-9)-5(3)}\left[\begin{array}{ll} -9 & -5 \\ -3 & -2 \end{array}\right] \\ & =\frac{1}{3}\left[\begin{array}{ll} -9 & -5 \\ -3 & -2 \end{array}\right] \\ & =\left[\begin{array}{rr} -3 & -\frac{5}{3} \\ -1 & -\frac{2}{3} \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} &\begin{aligned} & {\left[\begin{array}{cc} 4 & 2 \\ 2 & 1 \end{array}\right] } \\ & a d-b c=(4)(1)-2(2) \\ &=0 \end{aligned}\\ &\text { Maka matriks songsang tidak wujud } \end{aligned} $$
Soalan 2:
Hitung matriks songsang bagi matriks yang berikut.
Penyelesaian:
(a)
$$ \begin{aligned} &\left[\begin{array}{ll} 5 & 6 \\ 2 & 3 \end{array}\right]\\ &\begin{aligned} \text { Matriks songsang } & =\frac{1}{5(3)-6(2)}\left[\begin{array}{cc} 3 & -6 \\ -2 & 5 \end{array}\right] \\ & =\frac{1}{3}\left[\begin{array}{cc} 3 & -6 \\ -2 & 5 \end{array}\right] \\ & =\left[\begin{array}{cc} 1 & -2 \\ -\frac{2}{3} & \frac{5}{3} \end{array}\right] \end{aligned} \end{aligned} $$
(b)
$$ \begin{aligned} & {\left[\begin{array}{ll} 2 & 3 \\ 3 & 5 \end{array}\right]} \\ & \begin{aligned} \text { Matriks songsang } & =\frac{1}{2(5)-3(3)}\left[\begin{array}{cc} 5 & -3 \\ -3 & 2 \end{array}\right] \\ & =1\left[\begin{array}{cc} 5 & -3 \\ -3 & 2 \end{array}\right] \\ & =\left[\begin{array}{cc} 5 & -3 \\ -3 & 2 \end{array}\right] \end{aligned} \end{aligned} $$
(c)
$$ \begin{aligned} &\left[\begin{array}{cc} 4 & -2 \\ -3 & 2 \end{array}\right]\\ &\begin{aligned} \text { Matriks songsang } & =\frac{1}{4(2)-(-2)(-3)}\left[\begin{array}{ll} 2 & 2 \\ 3 & 4 \end{array}\right] \\ & =\frac{1}{2}\left[\begin{array}{ll} 2 & 2 \\ 3 & 4 \end{array}\right] \\ & =\left[\begin{array}{ll} 1 & 1 \\ \frac{3}{2} & 2 \end{array}\right] \end{aligned} \end{aligned} $$
(d)
$$ \begin{aligned} &\left[\begin{array}{cc} -2 & -5 \\ 2 & 7 \end{array}\right]\\ &\begin{aligned} \text { Matriks songsang } & =\frac{1}{(-2)(7)-(-5)(2)}\left[\begin{array}{cc} 7 & 5 \\ -2 & -2 \end{array}\right] \\ & =-\frac{1}{4}\left[\begin{array}{cc} 7 & 5 \\ -2 & -2 \end{array}\right] \\ & =\left[\begin{array}{cc} -\frac{7}{4} & -\frac{5}{4} \\ \frac{1}{2} & \frac{1}{2} \end{array}\right] \end{aligned} \end{aligned} $$
Hitung matriks songsang bagi matriks yang berikut.
Penyelesaian:
(a)
$$ \begin{aligned} &\left[\begin{array}{ll} 5 & 6 \\ 2 & 3 \end{array}\right]\\ &\begin{aligned} \text { Matriks songsang } & =\frac{1}{5(3)-6(2)}\left[\begin{array}{cc} 3 & -6 \\ -2 & 5 \end{array}\right] \\ & =\frac{1}{3}\left[\begin{array}{cc} 3 & -6 \\ -2 & 5 \end{array}\right] \\ & =\left[\begin{array}{cc} 1 & -2 \\ -\frac{2}{3} & \frac{5}{3} \end{array}\right] \end{aligned} \end{aligned} $$
(b)
$$ \begin{aligned} & {\left[\begin{array}{ll} 2 & 3 \\ 3 & 5 \end{array}\right]} \\ & \begin{aligned} \text { Matriks songsang } & =\frac{1}{2(5)-3(3)}\left[\begin{array}{cc} 5 & -3 \\ -3 & 2 \end{array}\right] \\ & =1\left[\begin{array}{cc} 5 & -3 \\ -3 & 2 \end{array}\right] \\ & =\left[\begin{array}{cc} 5 & -3 \\ -3 & 2 \end{array}\right] \end{aligned} \end{aligned} $$
(c)
$$ \begin{aligned} &\left[\begin{array}{cc} 4 & -2 \\ -3 & 2 \end{array}\right]\\ &\begin{aligned} \text { Matriks songsang } & =\frac{1}{4(2)-(-2)(-3)}\left[\begin{array}{ll} 2 & 2 \\ 3 & 4 \end{array}\right] \\ & =\frac{1}{2}\left[\begin{array}{ll} 2 & 2 \\ 3 & 4 \end{array}\right] \\ & =\left[\begin{array}{ll} 1 & 1 \\ \frac{3}{2} & 2 \end{array}\right] \end{aligned} \end{aligned} $$
(d)
$$ \begin{aligned} &\left[\begin{array}{cc} -2 & -5 \\ 2 & 7 \end{array}\right]\\ &\begin{aligned} \text { Matriks songsang } & =\frac{1}{(-2)(7)-(-5)(2)}\left[\begin{array}{cc} 7 & 5 \\ -2 & -2 \end{array}\right] \\ & =-\frac{1}{4}\left[\begin{array}{cc} 7 & 5 \\ -2 & -2 \end{array}\right] \\ & =\left[\begin{array}{cc} -\frac{7}{4} & -\frac{5}{4} \\ \frac{1}{2} & \frac{1}{2} \end{array}\right] \end{aligned} \end{aligned} $$
Soalan 3:
$$ \text { Diberi matriks } G=\left[\begin{array}{cc} 2 & 1 \\ 3 & p \end{array}\right] \text {. Hitung nilai } p \text { jika } $$
$$ \text { (a) matriks } G \text { tidak mempunyai matriks songsang, } $$
$$ \text { (b) } \quad G^{-1}=\left[\begin{array}{cc} \frac{4}{5} & -\frac{1}{5} \\ -\frac{3}{5} & \frac{2}{5} \end{array}\right] \text {. } $$
Penyelesaian:
(a)
$$ \begin{aligned} &\text { Diberi matriks } G \text { tidak mempunyai matriks songsang, }\\ &\begin{aligned} a d-b c & =0 \\ 2(p)-1(3) & =0 \\ 2 p-3 & =0 \\ 2 p & =3 \\ p & =\frac{3}{2} \end{aligned} \end{aligned} $$
(b)
$$ \begin{aligned} & G G^{-1}=\mathrm{I} \\ & {\left[\begin{array}{ll} 2 & 1 \\ 3 & p \end{array}\right]\left[\begin{array}{cc} \frac{4}{5} & -\frac{1}{5} \\ -\frac{3}{5} & \frac{2}{5} \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]} \\ & {\left[\begin{array}{ll} 2\left(\frac{4}{5}\right)+1\left(-\frac{3}{5}\right) & 2\left(-\frac{1}{5}\right)+1\left(\frac{2}{5}\right) \\ 3\left(\frac{4}{5}\right)+p\left(-\frac{3}{5}\right) & 3\left(-\frac{1}{5}\right)+p\left(\frac{2}{5}\right) \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]} \\ & {\left[\begin{array}{cc} 1 & 0 \\ \frac{12}{5}-\frac{3}{5} p & -\frac{3}{5}+\frac{2}{5} p \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]} \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} \frac{12}{5}-\frac{3}{5} p & =0 \\ \frac{3}{5} p & =\frac{12}{5} \\ p & =\frac{12}{5} \times \frac{5}{3} \\ p & =4 \end{aligned} \end{aligned} $$
$$ \text { Diberi matriks } G=\left[\begin{array}{cc} 2 & 1 \\ 3 & p \end{array}\right] \text {. Hitung nilai } p \text { jika } $$
$$ \text { (a) matriks } G \text { tidak mempunyai matriks songsang, } $$
$$ \text { (b) } \quad G^{-1}=\left[\begin{array}{cc} \frac{4}{5} & -\frac{1}{5} \\ -\frac{3}{5} & \frac{2}{5} \end{array}\right] \text {. } $$
Penyelesaian:
(a)
$$ \begin{aligned} &\text { Diberi matriks } G \text { tidak mempunyai matriks songsang, }\\ &\begin{aligned} a d-b c & =0 \\ 2(p)-1(3) & =0 \\ 2 p-3 & =0 \\ 2 p & =3 \\ p & =\frac{3}{2} \end{aligned} \end{aligned} $$
(b)
$$ \begin{aligned} & G G^{-1}=\mathrm{I} \\ & {\left[\begin{array}{ll} 2 & 1 \\ 3 & p \end{array}\right]\left[\begin{array}{cc} \frac{4}{5} & -\frac{1}{5} \\ -\frac{3}{5} & \frac{2}{5} \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]} \\ & {\left[\begin{array}{ll} 2\left(\frac{4}{5}\right)+1\left(-\frac{3}{5}\right) & 2\left(-\frac{1}{5}\right)+1\left(\frac{2}{5}\right) \\ 3\left(\frac{4}{5}\right)+p\left(-\frac{3}{5}\right) & 3\left(-\frac{1}{5}\right)+p\left(\frac{2}{5}\right) \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]} \\ & {\left[\begin{array}{cc} 1 & 0 \\ \frac{12}{5}-\frac{3}{5} p & -\frac{3}{5}+\frac{2}{5} p \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]} \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} \frac{12}{5}-\frac{3}{5} p & =0 \\ \frac{3}{5} p & =\frac{12}{5} \\ p & =\frac{12}{5} \times \frac{5}{3} \\ p & =4 \end{aligned} \end{aligned} $$
Soalan 4:
$$ \text { Diberi }\left[\begin{array}{cc} 4 & 10 \\ \frac{1}{2} & 1 \end{array}\right] P=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$
$$ \text { dan matriks } P \text { berperingkat } 2 \times 2 \text {. Hitung matriks } P \text {. } $$
Penyelesaian:
$$ \left[\begin{array}{cc} 4 & 10 \\ \frac{1}{2} & 1 \end{array}\right] P=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$
$$ \begin{aligned} \text { Matriks songsang } & =\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & d \end{array}\right] \\ P & =\frac{1}{4(1)-10\left(\frac{1}{2}\right)}\left[\begin{array}{cc} 1 & -10 \\ -\frac{1}{2} & 4 \end{array}\right] \\ P & =-1\left[\begin{array}{cc} 1 & -10 \\ -\frac{1}{2} & 4 \end{array}\right] \\ P & =\left[\begin{array}{cc} -1 & 10 \\ \frac{1}{2} & -4 \end{array}\right] \end{aligned} $$
$$ \text { Diberi }\left[\begin{array}{cc} 4 & 10 \\ \frac{1}{2} & 1 \end{array}\right] P=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$
$$ \text { dan matriks } P \text { berperingkat } 2 \times 2 \text {. Hitung matriks } P \text {. } $$
Penyelesaian:
$$ \left[\begin{array}{cc} 4 & 10 \\ \frac{1}{2} & 1 \end{array}\right] P=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$
$$ \begin{aligned} \text { Matriks songsang } & =\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & d \end{array}\right] \\ P & =\frac{1}{4(1)-10\left(\frac{1}{2}\right)}\left[\begin{array}{cc} 1 & -10 \\ -\frac{1}{2} & 4 \end{array}\right] \\ P & =-1\left[\begin{array}{cc} 1 & -10 \\ -\frac{1}{2} & 4 \end{array}\right] \\ P & =\left[\begin{array}{cc} -1 & 10 \\ \frac{1}{2} & -4 \end{array}\right] \end{aligned} $$