Add Question 7

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Ceda EI 2020-05-05 05:49:09 +05:30
parent c8f8f72d59
commit 7790cf75aa
1 changed files with 94 additions and 15 deletions

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