diff --git a/maths/assignment-matrices/assign.tex b/maths/assignment-matrices/assign.tex index 244185d..080add6 100644 --- a/maths/assignment-matrices/assign.tex +++ b/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}