Perkalian Skalar dengan Matriks
Berikut ini adalah cara mengalikan skalar \(k\) dengan matriks \(\text{A}\). \(\color{blue}k\cdot \begin{bmatrix}a_{11} & a_{12} \\a_{21} & a_{22} \end{bmatrix} = \color{blue}\begin{bmatrix}ka_{11} & ka_{12} \\ka_{21} & ka_{22} \end{bmatrix}\) Contoh: \(2\cdot \begin{bmatrix}-1 & 0 \\3 & 2\end{bmatrix} = \begin{bmatrix}2(-1) & 2(0) \\2(3) & 2(2)\end{bmatrix}\) \(2\cdot \begin{bmatrix}-1 & 0 \\3 & 2\end{bmatrix} = \begin{bmatrix}-2 & 0\\6& 4\end{bmatrix}\)Perkalian Matriks dengan Matriks
Matriks A dan Matriks B dapat dikalikan jika jumlah kolom pada matriks A sama dengan jumlah baris pada matriks B. \(\textbf{A} = \color{blue}\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\end{bmatrix}_{2 \times \color{red}3}\) \(\textbf{B} = \color{blue}\begin{bmatrix}b_{11} & b_{12} \\b_{21} & b_{22}\\b_{31}&b_{32}\end{bmatrix}_{\color{red}3\color{black} \times 2}\) \(\textbf{A}_{2 \times \color{red} \cancel{3}}\times \textbf{B}_{\color{red}\cancel{3}\color{black} \times 2} = \textbf{C}_{2 \times 2}\)
$$\textbf{C} = \color{blue}\begin{bmatrix}a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} & a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32}\\a_{21}b_{11}+a_{22}b_{21} + a_{23}b_{31} & a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32}\end{bmatrix}_{2 \times 2}$$
Contoh 1
\(\textbf{A} = \begin{bmatrix}1 & 0 & 3\\-3 & 2 & 5\end{bmatrix}\)
\(\textbf{B} = \begin{bmatrix}-2 & 1 \\0 &3\\4&-5\end{bmatrix}\)
\(\textbf{C}_{2 \times 2} = \textbf{A}_{2\times \color{red} \cancel{3}} \times \textbf{B}_{\color{red} \cancel{3}\color{black} \times 2}\)
$\textbf{C} = \begin{bmatrix}1(-2) + 0 + 3(4) & 1(1) + 0(3) + 3(-5)\\-3(-2)+2(0) + 5(4) & -3(1) +2(3) + 5(-5)\end{bmatrix}$
$\textbf{C} = \begin{bmatrix}-2 + 0 + 12 & 1+ 0\: -\: 15\\6+0 + 20 & -3 +6 \:-\:25\end{bmatrix}$
\(\textbf{C} = \begin{bmatrix}10 & -14\\26 & -22\end{bmatrix}\)
Contoh 2
\(\textbf{P} = \begin{bmatrix}1 & 0\\2 & 1 \\ 0 & 1\end{bmatrix}\)
\(\textbf{Q} = \begin{bmatrix}4 & 1 & 3\\0 &1 & 2\end{bmatrix}\)
\(\textbf{R}_{3\times 3} = \textbf{P}_{3 \times \color{red} \cancel{2}} \times \textbf{Q}_{\color{red}\cancel{2}\color{black} \times 3}\)
$$\textbf{R} = \begin{bmatrix}1(4) + 0 & 1(1) + 0 & 1(3) + 0(2)\\2(4) + 1(0) & 2(1) + 1(1)&2(3) + 1(2)\\0(4) + 1(0) & 0(1) + 1(1)& 0(3) + 1(2)\end{bmatrix}$$
\(\textbf{R} = \begin{bmatrix}4&1&3\\8&3&8\\0&1&2\end{bmatrix}\)
Contoh 3
Jika \(\textbf{A} = \begin{bmatrix}\sqrt{2} & -\sqrt{2}\\ \sqrt{2} & \sqrt{2} \end{bmatrix}\), hasil \(\textbf{A}^{1110} \) adalah…
Penyelesaian:
Matriks A dapat juga ditulis sebagai berikut:
\(\textbf{A} = 2\begin{bmatrix}\cos 45^{\circ} & -\sin 45^{\circ}\\ \sin 45^{\circ} & \cos 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^2 = \textbf{A} \cdot \textbf{A}\)
\(\textbf{A}^2 = 2\begin{bmatrix}\cos 45^{\circ} & -\sin 45^{\circ}\\ \sin 45^{\circ} & \cos 45^{\circ} \end{bmatrix} \cdot 2\begin{bmatrix}\cos 45^{\circ} & -\sin 45^{\circ}\\ \sin 45^{\circ} & \cos 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^2 = 2^2\begin{bmatrix}\cos^2 45^{\circ}\:-\: \sin^2 45^{\circ} & -2 \sin 45^{\circ} \cos 45^{\circ} \\ 2 \sin 45^{\circ} \cos 45^{\circ} & \cos^2 45^{\circ}\:-\:\sin^2 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^2 = 2^2\begin{bmatrix}\cos 90^{\circ} & -\sin 90^{\circ} \\ \sin 90^{\circ} & \cos 90^{\circ} \end{bmatrix}\)
\(\textbf{A}^2 = 2^2\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\)
\(\textbf{A}^3 = \textbf{A}^2 \cdot \textbf{A}\)
\(\textbf{A}^3 = 2^2\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix} \cdot 2\begin{bmatrix}\cos 45^{\circ} & -\sin 45^{\circ}\\ \sin 45^{\circ} & \cos 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^3 = 2^3 \begin{bmatrix}-\sin 45^{\circ} & -\cos 45^{\circ} \\ \cos 45^{\circ} & -\sin 45^{\circ} \end{bmatrix} \)
\(\textbf{A}^4 = \textbf{A}^3 \cdot \textbf{A}\)
\(\textbf{A}^4 = 2^3 \begin{bmatrix}-\sin 45^{\circ} & -\cos 45^{\circ} \\ \cos 45^{\circ} & -\sin 45^{\circ} \end{bmatrix} \cdot 2\begin{bmatrix}\cos 45^{\circ} & -\sin 45^{\circ}\\ \sin 45^{\circ} & \cos 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^4 = 2^4 \begin{bmatrix} -2 \sin 45^{\circ} \cos 45^{\circ} & \sin^2 45^{\circ} \:-\: \cos^2 45^{\circ} \\ \cos^2 45^{\circ} \:-\: \sin^2 45^{\circ} & -2\sin 45^{\circ} \cos 45^{\circ}\end{bmatrix}\)
\(\textbf{A}^4 = 2^4 \begin{bmatrix} -\sin 90^{\circ} & -\cos 90^{\circ} \\ \cos 90^{\circ} & -\sin 90^{\circ} \end{bmatrix}\)
\(\textbf{A}^4 = 2^4 \begin{bmatrix} -1 & 0 \\ 0 & -1\end{bmatrix}\)
\(\textbf{A}^5 = \textbf{A}^4 \cdot \textbf{A}\)
\(\textbf{A}^5 = 2^4 \begin{bmatrix} -1 & 0 \\ 0 & -1\end{bmatrix} \cdot 2\begin{bmatrix}\cos 45^{\circ} & -\sin 45^{\circ}\\ \sin 45^{\circ} & \cos 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^5 = 2^5 \begin{bmatrix} -\cos 45^{\circ} & \sin 45^{\circ} \\ -\sin 45^{\circ} & -\cos 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^6 = \textbf{A}^5 \cdot \textbf{A}\)
\(\textbf{A}^6 = 2^5 \begin{bmatrix} -\cos 45^{\circ} & \sin 45^{\circ} \\ -\sin 45^{\circ} & -\cos 45^{\circ} \end{bmatrix} \cdot 2\begin{bmatrix}\cos 45^{\circ} & -\sin 45^{\circ}\\ \sin 45^{\circ} & \cos 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^6 = 2^6 \begin{bmatrix} \sin^2 45^{\circ} \:-\: \cos^2 45^{\circ} & 2 \sin 45^{\circ} \cos 45^{\circ} \\ -2 \sin 45^{\circ} \cos 45^{\circ} & \sin^2 45^{\circ} \:-\: \cos^2 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^6 = 2^6 \begin{bmatrix} -\cos 90^{\circ} & \sin 90^{\circ} \\ -\sin 90^{\circ} & -\cos 90^{\circ} \end{bmatrix}\)
\(\textbf{A}^6 = 2^6 \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)
\(\textbf{A}^7 = \textbf{A}^6 \cdot \textbf{A}\)
\(\textbf{A}^7 = 2^6 \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \cdot 2\begin{bmatrix}\cos 45^{\circ} & -\sin 45^{\circ}\\ \sin 45^{\circ} & \cos 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^7 = 2^7 \begin{bmatrix} \sin 45^{\circ} & \cos 45^{\circ} \\ -\cos 45^{\circ} & \sin 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^8 = \textbf{A}^7 \cdot \textbf{A}\)
\(\textbf{A}^8 = 2^7 \begin{bmatrix} \sin 45^{\circ} & \cos 45^{\circ} \\ -\cos 45^{\circ} & \sin 45^{\circ} \end{bmatrix} \cdot 2\begin{bmatrix}\cos 45^{\circ} & -\sin 45^{\circ}\\ \sin 45^{\circ} & \cos 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^8 = 2^8 \begin{bmatrix} 2 \sin 45^{\circ} \cos 45^{\circ} & \cos^2 45^{\circ} \:-\: \sin^2 45^{\circ} \\ \sin^2 45^{\circ} \:-\: \cos^2 45^{\circ} & 2 \sin 45^{\circ} \cos 45^{\circ} \end{bmatrix}\)
\(\textbf{A}^8 = 2^8 \begin{bmatrix} \sin 90^{\circ} & \cos 90^{\circ} \\ -\cos 90^{\circ} & \sin 90^{\circ} \end{bmatrix}\)
\(\textbf{A}^8 = 2^8 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
\(\textbf{A}^8 = 2^8 \cdot \textbf{I}\) dengan \(\textbf{I}\) adalah matriks identitas berukuran 2 × 2
\(\textbf{A}^{1110} = \textbf{A}^{8(138) + 6}\)
\(\textbf{A}^{1110} = (\textbf{A}^{8})^{138} \cdot \textbf{A}^{6}\)
\(\textbf{A}^{1110} = (2^8 \cdot \textbf{I})^{138} \cdot 2^6 \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)
\(\textbf{A}^{1110} = 2^{1104} \cdot 2^6 \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)
\(\color{blue} \textbf{A}^{1110} = 2^{1110} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)