Add Question 7

maths/assignment-matrices/assign.tex

@@ -212,21 +212,23 @@ $, find $ adj. A$
 Adjoint of a matrix is the transpose of the cofactor matrix of the original matrix
 
 \[
-	A_{11} =
+	Cofactor\;of\;A_{11} =
 	\begin{vmatrix}
 		4 & 5 \\
 		-6 & -7
 	\end{vmatrix}
 	= 2
-	\;\;
+\]
-	A_{12} =
+\[
+	Cofactor\;of\;A_{12} =
 	\begin{vmatrix}
 		3 & 5 \\
 		0 & -7
 	\end{vmatrix}
 	= -21
-	\;\;
+\]
-	A_{13} =
+\[
+	Cofactor\;of\;A_{13} =
 	\begin{vmatrix}
 		3 & 4 \\
 		0 & -6
@@ -234,21 +236,23 @@ Adjoint of a matrix is the transpose of the cofactor matrix of the original matr
 	= -18
 \]
 \[
-	A_{21} =
+	Cofactor\;of\;A_{21} =
 	\begin{vmatrix}
 		0 & -1 \\
 		-6 & -7
 	\end{vmatrix}
 	= -6
-	\;\;
+\]
-	A_{22} =
+\[
+	Cofactor\;of\;A_{22} =
 	\begin{vmatrix}
 		1 & -1 \\
 		0 & -7
 	\end{vmatrix}
 	= -7
-	\;\;
+\]
-	A_{23} =
+\[
+	Cofactor\;of\;A_{23} =
 	\begin{vmatrix}
 		1 & -1 \\
 		0 & -6
@@ -256,21 +260,23 @@ Adjoint of a matrix is the transpose of the cofactor matrix of the original matr
 	= -6
 \]
 \[
-	A_{31} =
+	Cofactor\;of\;A_{31} =
 	\begin{vmatrix}
 		0 & -1 \\
 		4 & 5
 	\end{vmatrix}
 	= 4
-	\;\;
+\]
-	A_{32} =
+\[
+	Cofactor\;of\;A_{32} =
 	\begin{vmatrix}
 		1 & -1 \\
 		3 & 5
 	\end{vmatrix}
 	= 8
-	\;\;
+\]
-	A_{33} =
+\[
+	Cofactor\;of\;A_{33} =
 	\begin{vmatrix}
 		1 & 0 \\
 		3 & 4
@@ -297,4 +303,77 @@ Adjoint matrix is the transpose of Cofactor Matrix.
 	\end{bmatrix}
 \]
 
+\section{Question 7}
+
+\( A =
+\begin{bmatrix}
+	0 & 0 & 1 \\
+	0 & 1 & 0 \\
+	1 & 0 & 0
+\end{bmatrix}
+\), show that $A^{-1} = A$.
+
+We know that
+\[
+	A^{-1} = \frac{adj.(A)}{|A|}
+\]
+
+\begin{align*}
+	Cofactor\;of\;A_{11} &= 0 \\
+	Cofactor\;of\;A_{12} &= 0 \\
+	Cofactor\;of\;A_{13} &= -1 \\
+	Cofactor\;of\;A_{21} &= 0 \\
+	Cofactor\;of\;A_{22} &= -1 \\
+	Cofactor\;of\;A_{23} &= 0 \\
+	Cofactor\;of\;A_{31} &= -1 \\
+	Cofactor\;of\;A_{32} &= 0 \\
+	Cofactor\;of\;A_{33} &= 0
+\end{align*}
+
+\[
+	Cofactor\;Matrix =
+	\begin{bmatrix}
+		0 & 0 & -1 \\
+		0 & -1 & 0 \\
+		-1 & 0 & 0
+	\end{bmatrix}
+\]
+\[
+	adj.A =
+	\begin{bmatrix}
+		0 & 0 & -1 \\
+		0 & -1 & 0 \\
+		-1 & 0 & 0
+	\end{bmatrix}
+\]
+
+\[
+	|A| =
+	\begin{vmatrix}
+		0 & 0 & 1 \\
+		0 & 1 & 0 \\
+		1 & 0 & 0
+	\end{vmatrix}
+	 = 1 \times
+	 \begin{vmatrix}
+		 0 & 1 \\
+		 1 & 0
+	 \end{vmatrix}
+	 = 1 \times -1 = -1
+ \]
+
+\[
+	A^{-1} = \frac{adj.A}{|A|}
+\]
+\[
+	A^{-1} =
+	\begin{bmatrix}
+		0 & 0 & 1 \\
+		0 & 1 & 0 \\
+		1 & 0 & 0
+	\end{bmatrix}
+\]
+\[
+	\therefore A^{-1} = A
+\]
 \end{document}