Add Question 6

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Ceda EI 2020-05-05 05:32:14 +05:30
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@ -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}