Add Question 6

maths/assignment-matrices/assign.tex

@@ -9,7 +9,7 @@
 \date{05 May, 2020}
 \maketitle
 \section{Question 1}
-\begin{equation}
+\[
 	A =
 	\begin{bmatrix}
 		2 & -5 & -1 \\
@@ -20,42 +20,42 @@
 		3 & 4 & 0 \\
 		5 & -2 & 3
 	\end{bmatrix}
-\end{equation}
+\]
 
 \subsection{Part i}
-\begin{equation}
+\[
 	A + B =
 	\begin{bmatrix}
 		2 + 1 & -5 + 4 & -1 + 0 \\
 		-2 + 5 & -1 -2 & 4 + 3
 	\end{bmatrix}
-\end{equation}
+\]
-\begin{equation}
+\[
 	A + B =
 	\begin{bmatrix}
 		3 & -1 & -1 \\
 		3 & -3 & 7
 	\end{bmatrix}
-\end{equation}
+\]
 \subsection{Part ii}
-\begin{equation}
+\[
 	2A + B =
 	\begin{bmatrix}
 		4 + 1 & -10 + 4 & -2 + 0 \\
 		-4 + 5 & -2 -2 & 8 + 3
 	\end{bmatrix}
-\end{equation}
+\]
-\begin{equation}
+\[
 	2A + B =
 	\begin{bmatrix}
 		5 & -6 & -2 \\
 		1 & -4 & 11
 	\end{bmatrix}
-\end{equation}
+\]
 
 
 \section{Question 2}
-\begin{equation}
+\[
 	A =
 	\begin{bmatrix}
 		1 & 2 & 3 \\
@@ -68,8 +68,8 @@
 		2 & 1 & 2 \\
 		5 & 2 & 3
 	\end{bmatrix}
-\end{equation}
+\]
-\begin{equation}
+\[
 	AB =
 	\begin{bmatrix}
 		1 & 2 & 3 \\
@@ -81,23 +81,23 @@
 		2 & 1 & 2 \\
 		5 & 2 & 3
 	\end{bmatrix}
-\end{equation}
+\]
-\begin{equation}
+\[
 	AB =
 	\begin{bmatrix}
 		1 \times 1 + 2 \times 2 + 3 \times 5 & 1 \times 0 + 2 \times 1 + 3 \times 2 & 1 \times 2 + 2 \times 2 + 3 \times 3 \\
 		4 \times 1 + 5 \times 2 + 6 \times 5 & 4 \times 0 + 5 \times 1 + 6 \times 2 & 4 \times 2 + 5 \times 2 + 6 \times 3 \\
 		7 \times 1 + 8 \times 2 + 9 \times 5 & 7 \times 0 + 8 \times 1 + 9 \times 2 & 7 \times 2 + 8 \times 2 + 9 \times 3
 	\end{bmatrix}
-\end{equation}
+\]
-\begin{equation}
+\[
 	AB =
 	\begin{bmatrix}
 		20 & 8 & 16 \\
 		44 & 17 & 36 \\
 		68 & 26 & 57
 	\end{bmatrix}
-\end{equation}
+\]
 
 
 \section{Question 3}
@@ -198,4 +198,103 @@ A =
 \[
 	\therefore {(A^{\theta})}^{\theta} = A
 \]
+
+\section{Question 6}
+
+$ A =
+\begin{bmatrix}
+	1 & 0 & -1 \\
+	3 & 4 & 5 \\
+	0 & -6 & -7
+\end{bmatrix}
+$, find $ adj. A$
+
+Adjoint of a matrix is the transpose of the cofactor matrix of the original matrix
+
+\[
+	A_{11} =
+	\begin{vmatrix}
+		4 & 5 \\
+		-6 & -7
+	\end{vmatrix}
+	= 2
+	\;\;
+	A_{12} =
+	\begin{vmatrix}
+		3 & 5 \\
+		0 & -7
+	\end{vmatrix}
+	= -21
+	\;\;
+	A_{13} =
+	\begin{vmatrix}
+		3 & 4 \\
+		0 & -6
+	\end{vmatrix}
+	= -18
+\]
+\[
+	A_{21} =
+	\begin{vmatrix}
+		0 & -1 \\
+		-6 & -7
+	\end{vmatrix}
+	= -6
+	\;\;
+	A_{22} =
+	\begin{vmatrix}
+		1 & -1 \\
+		0 & -7
+	\end{vmatrix}
+	= -7
+	\;\;
+	A_{23} =
+	\begin{vmatrix}
+		1 & -1 \\
+		0 & -6
+	\end{vmatrix}
+	= -6
+\]
+\[
+	A_{31} =
+	\begin{vmatrix}
+		0 & -1 \\
+		4 & 5
+	\end{vmatrix}
+	= 4
+	\;\;
+	A_{32} =
+	\begin{vmatrix}
+		1 & -1 \\
+		3 & 5
+	\end{vmatrix}
+	= 8
+	\;\;
+	A_{33} =
+	\begin{vmatrix}
+		1 & 0 \\
+		3 & 4
+	\end{vmatrix}
+	= 4
+\]
+
+\[
+	Cofactor\;Matrix =
+	\begin{bmatrix}
+		2 & -21 & -18 \\
+		-6 & -7 & -6 \\
+		4 & 8 & 4
+	\end{bmatrix}
+\]
+
+Adjoint matrix is the transpose of Cofactor Matrix.
+\[
+	\therefore adj.A =
+	\begin{bmatrix}
+		2 & -6 & 4 \\
+		-21 & -7 & 8 \\
+		-18 & -6 & 4
+	\end{bmatrix}
+\]
+
 \end{document}