<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\un}{\underline}
\parindent=0pt
\begin{document}

{\bf Question}

Find the inverses of the following matrices and verify that they
are correct.

\begin{description}
\item[(i)]
$A=\left(\begin{array}{cc} 0 &amp; 1\\ 1 &amp; 0 \end{array}\right)$

\item[(ii)]
$B=\left(\begin{array}{cc} 1 &amp; 1\\ 1 &amp; 0 \end{array}\right)$

\item[(iii)]
$C=\left(\begin{array}{cc} 2 &amp; -2\\ 6 &amp; -3 \end{array}\right)$
\end{description}

\medskip

{\bf Answer}
\begin{description}
\item[(i)]
$A=\left(\begin{array}{cc}0 &amp; 1\\1 &amp; 0 \end{array} \right)
\Rightarrow det A=0-1=-1$

and $A^{-1}=\ds\frac{1}{-1}\left(\begin{array}{cc}0 &amp; -1\\ -1 &amp; 0
\end{array} \right)=\left(\begin{array}{cc}0 &amp; 1\\1 &amp; 0
\end{array}\right)$

i.e., $A^{-1}=A$!!!

Check $A^{-1}A=I_2=\left(\begin{array}{cc}1 &amp; 0\\ 0 &amp; 1
\end{array}\right)$

$A^{-1}A=\left(\begin{array}{cc}0 &amp; 1\\1 &amp; 0 \end{array}
\right)\left(\begin{array}{cc}0 &amp; 1\\1 &amp; 0 \end{array}
\right)=\left(\begin{array}{cc}1 &amp; 0\\ 0 &amp; 1
\end{array}\right)\ \surd$

Should also check $AA^{-1}=I_2$ (obvious here)
\item[(ii)]
$B=\left(\begin{array}{cc} 1 &amp; 1\\ 1 &amp; 0 \end{array} \right)
\Rightarrow det B = 1 \times 0 - 1 \times 1=-1$

and $B^{-1}=\ds\frac{1}{-1}\left(\begin{array}{cc}0 &amp; -1\\ -1 &amp; 1
\end{array} \right)=\left(\begin{array}{cc}0 &amp; 1\\1 &amp; -1
\end{array}\right)$

Check $B^{-1}B=\left(\begin{array}{cc}0 &amp; 1\\1 &amp; -1
\end{array}\right)\left(\begin{array}{cc} 1 &amp; 1\\ 1 &amp; 0
\end{array} \right)=\left(\begin{array}{cc}1 &amp; 0\\ 0 &amp; 1
\end{array}\right)=I_2\ \surd$

$BB^{-1}=\left(\begin{array}{cc}1 &amp; 1\\1 &amp; 0
\end{array}\right)\left(\begin{array}{cc} 0 &amp; 1\\ 1 &amp; -1
\end{array} \right)=\left(\begin{array}{cc}1 &amp; 0\\ 0 &amp; 1
\end{array}\right)=I_2\ \surd$

\newpage
\item[(iii)]
$C=\left(\begin{array}{cc} 2 &amp; -2\\ 6 &amp; -3 \end{array} \right)
\Rightarrow det B = -6-(-12)=6$

and $C^{-1}=\ds\frac{1}{6}\left(\begin{array}{cc}-3 &amp; 2\\ -6 &amp; 2
\end{array} \right)=\left(\begin{array}{cc}-\frac{1}{2} &amp; \frac{1}{3}\\-1 &amp;
\frac{1}{3} \end{array}\right)$

Check $C^{-1}C=\left(\begin{array}{cc}-\frac{1}{2} &amp;
\frac{1}{3}\\-1 &amp; \frac{1}{3}
\end{array}\right)\left(\begin{array}{cc} 2 &amp; -2\\ 6 &amp; -3
\end{array} \right)=\left(\begin{array}{cc}1 &amp; 0\\ 0 &amp; 1
\end{array}\right)=I_2\ \surd$

$CC^{-1}=\left(\begin{array}{cc}2 &amp; -2\\ 6 &amp; -3
\end{array}\right)\left(\begin{array}{cc} -\frac{1}{2} &amp;
\frac{1}{3}\\-1 &amp; \frac{1}{3}
\end{array} \right)=\left(\begin{array}{cc}1 &amp; 0\\ 0 &amp; 1
\end{array}\right)=I_2\ \surd$

\end{description}
\end{document}
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