diff --git a/maths/assignment-matrices/assign.tex b/maths/assignment-matrices/assign.tex index 259f087..244185d 100644 --- a/maths/assignment-matrices/assign.tex +++ b/maths/assignment-matrices/assign.tex @@ -1,6 +1,8 @@ \documentclass{article} % Import for matrices \usepackage{amsmath} +% Import for therefore symbol +\usepackage{amssymb} \begin{document} \title{Mathematics Assignment --- Matrices} \author{Ahmad Saalim Lone, 2019BCSE017} @@ -50,6 +52,8 @@ 1 & -4 & 11 \end{bmatrix} \end{equation} + + \section{Question 2} \begin{equation} A = @@ -81,9 +85,9 @@ \begin{equation} AB = \begin{bmatrix} - 1 * 1 + 2 * 2 + 3 * 5 & 1 * 0 + 2 * 1 + 3 * 2 & 1 * 2 + 2 * 2 + 3 * 3 \\ - 4 * 1 + 5 * 2 + 6 * 5 & 4 * 0 + 5 * 1 + 6 * 2 & 4 * 2 + 5 * 2 + 6 * 3 \\ - 7 * 1 + 8 * 2 + 9 * 5 & 7 * 0 + 8 * 1 + 9 * 2 & 7 * 2 + 8 * 2 + 9 * 3 + 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} @@ -94,4 +98,104 @@ 68 & 26 & 57 \end{bmatrix} \end{equation} + + +\section{Question 3} +If \( +A = +\begin{bmatrix} + 1 & -2 & -3 \\ + -4 & 2 & 5 +\end{bmatrix} +B = +\begin{bmatrix} + 2 & 3 \\ + 4 & 5 \\ + 2 & 1 +\end{bmatrix} +\), show that \(AB \ne BA\). + +Order of $A$ = $2\times3$ + +Order of $B$ = $3\times2$ + +Order of $AB$ = $rows \; of \; A \times columns \; of \; B$ = $2\times2$ + +Order of $BA$ = $rows \; of \; B \times columns \; of \; A$ = $3\times3$ + +Matrices of different order can't be equal. + +$\therefore AB \ne BA$ + +\section{Question 4} + +Show that \( +A = +\begin{bmatrix} + 3 & 1 + 2i \\ + 1-2i & 2 +\end{bmatrix} +\) is a hermitian. + +For a matrix to be hermitian, each element $a_{i,j}$ needs to be the complex +conjugate of the element at $a_{j,i}$. In given matrix, we have + +\begin{itemize} + \item \(a_{11} = 3\) + \item \(a_{12} = 1 + 2i\) + \item \(a_{21} = 1 - 2i\) + \item \(a_{22} = 2\) +\end{itemize} + +The conjugates are as follows + +\begin{itemize} + \item \(\overline{a_{11}} = 3\) + \item \(\overline{a_{12}} = 1 - 2i\) + \item \(\overline{a_{21}} = 1 + 2i\) + \item \(\overline{a_{22}} = 2\) +\end{itemize} + +As we can see, \(\overline{a_{11}} = a_{11}\), \(\overline{a_{12}} = a_{21}\), \(\overline{a_{21}} = a_{12}\) and \(\overline{a_{22}} = a_{22}\). + +$\therefore A$ is hermitian. + +\section{Question 5} + +If \( +A = +\begin{bmatrix} + 5 & 1 + i \\ + -1 + i & 4 +\end{bmatrix} +\), show that ${(A^{\theta})}^{\theta}$ + +\[ + A = + \begin{bmatrix} + 5 & 1 + i \\ + -1 + i & 4 + \end{bmatrix} +\] + +\[ + \overline{A^{\theta}} = + \begin{bmatrix} + 5 & - 1 - i \\ + 1 + i & 4 + \end{bmatrix} +\] + + +\[ + {(A^{\theta})}^{\theta} = + \begin{bmatrix} + 5 & 1 + i \\ + -1 + i & 4 + \end{bmatrix} +\] + +\[ + \therefore {(A^{\theta})}^{\theta} = A +\] \end{document}