Contoh Soal 2 Matriks

$\begin{array}{l}\\ 6.& \text{Diketahui matriks}\\ & \text{M}=\left(\begin{array}{ccc}-6& 9& -15\\ 3& -6& 12\end{array}\right)\\ & \text{dan}\text{N}=\left(\begin{array}{ccc}2& -3& 5\\ -1& 2& -4\end{array}\right).\\ & \text{Nilai}k\text{yang memenuhi jika}\\ & \text{M}=k\text{N}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& -{\displaystyle \frac{1}{3}}\\ \text{b}.& {\displaystyle \frac{1}{3}}\\ \text{c}.& -1\\ \text{d}.& -3\\ \text{e}.& 3\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{d}}\\ & \begin{array}{rl}& \text{Diketahu bahwa}\\ & \text{M}=k\text{N}\\ & (\text{perkalian suatu matrik dengan skalar})\\ & \left(\begin{array}{ccc}-6& 9& -15\\ 3& -6& 12\end{array}\right)\\ & =\left(\begin{array}{ccc}-3.2& -3.-3& -3.5\\ -3.-1& -3.2& -3.-4\end{array}\right)\\ & =-3\left(\begin{array}{ccc}2& -3& 5\\ -1& 2& -4\end{array}\right)\\ & =k\left(\begin{array}{ccc}2& -3& 5\\ -1& 2& -4\end{array}\right)\\ & \text{sehingga dari kesamaan tersebut}\\ & \text{maka}k=-3\end{array}\end{array}$

$\begin{array}{l}\\ 7.& \text{Hasil dari}\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)\times \left(\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right)\\ & \text{adalah}...\\ & \begin{array}{l}\\ \text{a}.& \left(\begin{array}{cc}22& 28\\ 49& 64\end{array}\right)\\ \text{b}.& \left(\begin{array}{cc}22& 49\\ 28& 64\end{array}\right)\\ \text{c}.& \left(\begin{array}{cc}64& 28\\ 49& 22\end{array}\right)\\ \text{d}.& \left(\begin{array}{ccc}2& 8& 18\\ 4& 15& 30\end{array}\right)\\ \text{e}.& \left(\begin{array}{ccc}1& 4& 6\\ 4& 15& 30\end{array}\right)\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{a}}\\ & \begin{array}{rl}& {\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)}_{2\times 3}\times {\left(\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right)}_{3\times 2}\\ & ={\left(\begin{array}{cc}1.1+2.3+3.5& 1.2+2.4+3.6\\ 4.1+5.3+6.5& 4.2+5.4+6.6\end{array}\right)}_{2\times 2}\\ & ={\left(\begin{array}{cc}1+6+15& 2+8+18\\ 4+15+30& 8+20+36\end{array}\right)}_{2\times 2}\\ & ={\left(\begin{array}{cc}22& 28\\ 49& 64\end{array}\right)}_{2\times 2}\end{array}\end{array}$

$\begin{array}{l}\\ 8.& \text{Jika diketahui matriks}\\ & \text{A}=\left(\begin{array}{cc}0& 1\\ 3& 2\end{array}\right).\\ & \text{maka hasil dari}{\text{A}}^3\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& \left(\begin{array}{cc}5& 8\\ 20& 22\end{array}\right)\\ \text{b}.& \left(\begin{array}{cc}6& 7\\ 21& 20\end{array}\right)\\ \text{c}.& \left(\begin{array}{cc}6& 7\\ 20& 22\end{array}\right)\\ \text{d}.& \left(\begin{array}{cc}7& 8\\ 20& 23\end{array}\right)\\ \text{e}.& \left(\begin{array}{cc}7& 9\\ 20& 23\end{array}\right)\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{b}}\\ & \begin{array}{rl}\text{Dike}& \text{tahui bahwa}\\ \text{A}& =\left(\begin{array}{cc}0& 1\\ 3& 2\end{array}\right)\\ \text{mak}& \text{a}\\ {\text{A}}^2& =\text{A}\times \text{A}\\ & =\left(\begin{array}{cc}0& 1\\ 3& 2\end{array}\right)\times \left(\begin{array}{cc}0& 1\\ 3& 2\end{array}\right)\\ & =\left(\begin{array}{cc}0+3& 0+2\\ 0+6& 3+4\end{array}\right)\\ & =\left(\begin{array}{cc}3& 2\\ 6& 7\end{array}\right)\\ {\text{A}}^3& ={\text{A}}^2\times \text{A}\\ & =\left(\begin{array}{cc}3& 2\\ 6& 7\end{array}\right)\times \left(\begin{array}{cc}0& 1\\ 3& 2\end{array}\right)\\ & =\left(\begin{array}{cc}0+6& 3+4\\ 0+21& 6+14\end{array}\right)\\ & =\left(\begin{array}{cc}6& 7\\ 21& 20\end{array}\right)\end{array}\end{array}$

$\begin{array}{l}\\ 9.& (\mathbf{\text{SBMPTN Mat IPA 2014}})\\ & \text{Jika}\text{A}\text{adalah matriks yang berordo}\\ & 2\times 2\text{dan memenuhi}\\ & \left(\begin{array}{cc}x& 1\end{array}\right)\times \text{A}\times \left(\begin{array}{c}x\\ 1\end{array}\right)=x^2-5x+8,\\ & \text{maka matriks A yang mungkin adalah}....\\ & \begin{array}{l}\\ \text{a}.& \left(\begin{array}{cc}1& -5\\ 8& 0\end{array}\right)\\ \text{b}.& \left(\begin{array}{cc}1& 5\\ 8& 0\end{array}\right)\\ \text{c}.& \left(\begin{array}{cc}1& 8\\ -5& 0\end{array}\right)\\ \text{d}.& \left(\begin{array}{cc}1& 3\\ -8& 8\end{array}\right)\\ \text{e}.& \left(\begin{array}{cc}1& -3\\ 8& -8\end{array}\right)\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{d}}\\ & \begin{array}{rl}\left(\begin{array}{cc}x& 1\end{array}\right)\times \text{A}\times \left(\begin{array}{c}x\\ 1\end{array}\right)& =x^2-5x+8\\ \left(\begin{array}{cc}x& 1\end{array}\right)\times \left(\begin{array}{cc}p& q\\ r& s\end{array}\right)\times \left(\begin{array}{c}x\\ 1\end{array}\right)& =x^2-5x+8\\ \left(\begin{array}{cc}xp+r& xq+s\end{array}\right)\times \left(\begin{array}{c}x\\ 1\end{array}\right)& =x^2-5x+8\\ \left(\begin{array}{c}{x}^{2}p+xr+xq+s\end{array}\right)& =x^2-5x+8\\ p{x}^2+(q+r)x+s& =x^2-5x+8\end{array}\\ & \begin{array}{rl}& \{\begin{array}{ll}p& =1\\ q+r& =-5\\ s& =8\end{array}\Rightarrow \left(\begin{array}{cc}1& ...\\ ...& 8\end{array}\right)\\ & \text{Sehingga yang paling mungkin}\\ & \text{adalah}\left(\begin{array}{cc}1& 3\\ -8& 8\end{array}\right)\end{array}\end{array}$

$\begin{array}{l}\\ 10.& \text{Diketahui}\\ & \left(\begin{array}{cc}{}^x\log a& \log (2a-6)\\ \log (b-2)& 1\end{array}\right)=\left(\begin{array}{cc}\log b& 1\\ \log a& 1\end{array}\right)\\ & \text{maka nilai}x\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& 1\\ \text{b}.& 2\\ \text{c}.& 4\\ \text{d}.& 6\\ \text{e}.& 8\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{e}}\\ & \begin{array}{rl}& \left(\begin{array}{cc}{}^x\log a& \log (2a-6)\\ \log (b-2)& 1\end{array}\right)=\left(\begin{array}{cc}\log b& 1\\ \log a& 1\end{array}\right)\\ & \text{maka}\\ & \{\begin{array}{ll}{}^x\log a& =\log b.........(1)\\ \log (2a-6)& =1..............(2)\\ \log (b-2)& =\log a.........(3)\end{array}\\ & \text{Sehingga}\text{dari persamaan}(2)\\ & \text{akan didapatkan}\\ & \log (2a-6)=1=\log 10\\ & (2a-6)=10\\ & a=8...........................(4)\\ & \text{persamaan}(4)\text{ke persamaan}(3),\\ & \text{maka}\\ & \log (b-2)=\log a\\ & b-2=a=8\\ & b=10.................................(5)\\ & \text{Selanjutnya dari persamaan}(5)\\ & \text{akan diperoleh}\\ & {}^x\log a=\log b\\ & {}^x\log 8=\log 10=1\\ & x^1=8\\ & \iff x=8\end{array}\end{array}$