Soalan 6:
$$ \text { Diberi } G=\left[\begin{array}{ll} 4 & r \\ 1 & 2 \end{array}\right] \text { dan } H=\left[\begin{array}{cc} 1 & 3 \\ -1 & s \end{array}\right] \text {. } $$
$$ \text { Hitung nilai } r \text { dan nilai } s \text { jika } G H=H G \text {. } $$
Penyelesaian:
$$ \begin{aligned} G H & =H G \\ {\left[\begin{array}{ll} 4 & r \\ 1 & 2 \end{array}\right]\left[\begin{array}{cc} 1 & 3 \\ -1 & s \end{array}\right] } & =\left[\begin{array}{cc} 1 & 3 \\ -1 & s \end{array}\right]\left[\begin{array}{ll} 4 & r \\ 1 & 2 \end{array}\right] \\ {\left[\begin{array}{cc} 4(1)+r(-1) & 4(3)+r s \\ 1(1)+2(-1) & 1(3)+2 s \end{array}\right] } & =\left[\begin{array}{cc} 1(4)+3(1) & 1(r)+3(2) \\ -1(4)+s(1) & -1(r)+s(2) \end{array}\right] \\ {\left[\begin{array}{cc} 4-r & 12+r s \\ -1 & 3+2 s \end{array}\right] } & =\left[\begin{array}{cc} 7 & r+6 \\ -4+s & -r+2 s \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} 4-r & =7 \\ r & =4-7 \\ r & =-3 \end{aligned} \end{aligned} $$
$$ \begin{aligned} -1 & =-4+s \\ s & =-1+4 \\ s & =3 \end{aligned} $$
$$ \text { Diberi } G=\left[\begin{array}{ll} 4 & r \\ 1 & 2 \end{array}\right] \text { dan } H=\left[\begin{array}{cc} 1 & 3 \\ -1 & s \end{array}\right] \text {. } $$
$$ \text { Hitung nilai } r \text { dan nilai } s \text { jika } G H=H G \text {. } $$
Penyelesaian:
$$ \begin{aligned} G H & =H G \\ {\left[\begin{array}{ll} 4 & r \\ 1 & 2 \end{array}\right]\left[\begin{array}{cc} 1 & 3 \\ -1 & s \end{array}\right] } & =\left[\begin{array}{cc} 1 & 3 \\ -1 & s \end{array}\right]\left[\begin{array}{ll} 4 & r \\ 1 & 2 \end{array}\right] \\ {\left[\begin{array}{cc} 4(1)+r(-1) & 4(3)+r s \\ 1(1)+2(-1) & 1(3)+2 s \end{array}\right] } & =\left[\begin{array}{cc} 1(4)+3(1) & 1(r)+3(2) \\ -1(4)+s(1) & -1(r)+s(2) \end{array}\right] \\ {\left[\begin{array}{cc} 4-r & 12+r s \\ -1 & 3+2 s \end{array}\right] } & =\left[\begin{array}{cc} 7 & r+6 \\ -4+s & -r+2 s \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} 4-r & =7 \\ r & =4-7 \\ r & =-3 \end{aligned} \end{aligned} $$
$$ \begin{aligned} -1 & =-4+s \\ s & =-1+4 \\ s & =3 \end{aligned} $$
Soalan 7:
$$ \text { Diberi } A=\left[\begin{array}{cc} 4 & -3 \\ 2 & 1 \end{array}\right] \text { dan } A B=I \text {. Hitung matriks } B \text {. } $$
Penyelesaian:
$$ A=\left[\begin{array}{cc} 4 & -3 \\ 2 & 1 \end{array}\right] $$
Diberi AB = I
B ialah matriks songsang bagi A.
$$ \begin{aligned} \text { Matriks songsang } & =\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right] \\ B & =\frac{1}{4(1)-(-3)(2)}\left[\begin{array}{cc} 1 & 3 \\ -2 & 4 \end{array}\right] \\ & =\frac{1}{10}\left[\begin{array}{cc} 1 & 3 \\ -2 & 4 \end{array}\right] \\ & =\left[\begin{array}{cc} \frac{1}{10} & \frac{3}{10} \\ -\frac{1}{5} & \frac{2}{5} \end{array}\right] \end{aligned} $$
$$ \text { Diberi } A=\left[\begin{array}{cc} 4 & -3 \\ 2 & 1 \end{array}\right] \text { dan } A B=I \text {. Hitung matriks } B \text {. } $$
Penyelesaian:
$$ A=\left[\begin{array}{cc} 4 & -3 \\ 2 & 1 \end{array}\right] $$
Diberi AB = I
B ialah matriks songsang bagi A.
$$ \begin{aligned} \text { Matriks songsang } & =\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right] \\ B & =\frac{1}{4(1)-(-3)(2)}\left[\begin{array}{cc} 1 & 3 \\ -2 & 4 \end{array}\right] \\ & =\frac{1}{10}\left[\begin{array}{cc} 1 & 3 \\ -2 & 4 \end{array}\right] \\ & =\left[\begin{array}{cc} \frac{1}{10} & \frac{3}{10} \\ -\frac{1}{5} & \frac{2}{5} \end{array}\right] \end{aligned} $$
Soalan 8:
$$ \text { Diberi } P=\left[\begin{array}{cc} 10 & -5 \\ -2 & 1 \\ 0 & 2 z-3 \end{array}\right], Q=\left[\begin{array}{cc} 1 & y \\ 0.2 & -\frac{1}{3} \\ 8 & -1 \end{array}\right] \text {, } $$
$$ R=\left[\begin{array}{cc} 11 & -25 \\ y+6 x & -0.2 \\ 24 & 9 \end{array}\right] \text { dan } 0.8 P+3 Q=R, $$
$$ \text { hitung nilai } x \text {, nilai } y \text { dan nilai } z \text {. } $$
Penyelesaian:
$$ \begin{aligned} 0.8 P+3 Q & =R \\ 0.8\left[\begin{array}{cc} 10 & -5 \\ -2 & 1 \\ 0 & 2 z-3 \end{array}\right]+3\left[\begin{array}{cc} 1 & y \\ 0.2 & -\frac{1}{3} \\ 8 & -1 \end{array}\right] & =\left[\begin{array}{cc} 11 & -25 \\ y+6 x & -0.2 \\ 24 & 9 \end{array}\right] \\ {\left[\begin{array}{cc} 8 & -4 \\ -1.6 & 0.8 \\ 0 & 1.6 z-2.4 \end{array}\right]+\left[\begin{array}{cc} 3 & 3 y \\ 0.6 & -1 \\ 24 & -3 \end{array}\right] } & =\left[\begin{array}{cc} 11 & -25 \\ y+6 x & -0.2 \\ 24 & 9 \end{array}\right] \\ {\left[\begin{array}{cc} 11 & -4+3 y \\ -1 & -0.2 \\ 24 & 1.6 z-5.4 \end{array}\right] } & =\left[\begin{array}{cc} 11 & -25 \\ y+6 x & -0.2 \\ 24 & 9 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} -4+3 y & =-25 \\ 3 y & =-21 \\ y & =-7 \end{aligned} \end{aligned} $$
$$ \begin{aligned} -1 & =y+6 x \\ -1 & =-7+6 x \\ 6 & =6 x \\ x & =1 \end{aligned} $$
$$ \begin{aligned} 1.6 z-5.4 & =9 \\ 1.6 z & =14.4 \\ z & =9 \end{aligned} $$
$$ \therefore x=1, y=-7, z=9 $$
$$ \text { Diberi } P=\left[\begin{array}{cc} 10 & -5 \\ -2 & 1 \\ 0 & 2 z-3 \end{array}\right], Q=\left[\begin{array}{cc} 1 & y \\ 0.2 & -\frac{1}{3} \\ 8 & -1 \end{array}\right] \text {, } $$
$$ R=\left[\begin{array}{cc} 11 & -25 \\ y+6 x & -0.2 \\ 24 & 9 \end{array}\right] \text { dan } 0.8 P+3 Q=R, $$
$$ \text { hitung nilai } x \text {, nilai } y \text { dan nilai } z \text {. } $$
Penyelesaian:
$$ \begin{aligned} 0.8 P+3 Q & =R \\ 0.8\left[\begin{array}{cc} 10 & -5 \\ -2 & 1 \\ 0 & 2 z-3 \end{array}\right]+3\left[\begin{array}{cc} 1 & y \\ 0.2 & -\frac{1}{3} \\ 8 & -1 \end{array}\right] & =\left[\begin{array}{cc} 11 & -25 \\ y+6 x & -0.2 \\ 24 & 9 \end{array}\right] \\ {\left[\begin{array}{cc} 8 & -4 \\ -1.6 & 0.8 \\ 0 & 1.6 z-2.4 \end{array}\right]+\left[\begin{array}{cc} 3 & 3 y \\ 0.6 & -1 \\ 24 & -3 \end{array}\right] } & =\left[\begin{array}{cc} 11 & -25 \\ y+6 x & -0.2 \\ 24 & 9 \end{array}\right] \\ {\left[\begin{array}{cc} 11 & -4+3 y \\ -1 & -0.2 \\ 24 & 1.6 z-5.4 \end{array}\right] } & =\left[\begin{array}{cc} 11 & -25 \\ y+6 x & -0.2 \\ 24 & 9 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} &\text { Bandingkan unsur-unsur sepadan, }\\ &\begin{aligned} -4+3 y & =-25 \\ 3 y & =-21 \\ y & =-7 \end{aligned} \end{aligned} $$
$$ \begin{aligned} -1 & =y+6 x \\ -1 & =-7+6 x \\ 6 & =6 x \\ x & =1 \end{aligned} $$
$$ \begin{aligned} 1.6 z-5.4 & =9 \\ 1.6 z & =14.4 \\ z & =9 \end{aligned} $$
$$ \therefore x=1, y=-7, z=9 $$