High School Mathematics Extensions/Matrices/Problem Set/Solutions

Content HSME Matrices Recurrence Relations Problem Set Project Exercises Solutions Problem Set Solutions Definition Sheet Full Version

Matrices Problem Set

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1.

$\begin{pmatrix} 2&3\\ 3&5\end{pmatrix} \begin{pmatrix} ?&?&?&?\\ ?&?&?&?\end{pmatrix} =\begin{pmatrix} 28&94&70&102\\ 44&153&112&163\end{pmatrix}$
$\begin{pmatrix} 2&3\\ 3&5\end{pmatrix}^{-1} \begin{pmatrix} 2&3\\ 3&5\end{pmatrix} \begin{pmatrix} ?&?&?&?\\ ?&?&?&?\end{pmatrix} =\begin{pmatrix} 2&3\\ 3&5\end{pmatrix}^{-1} \begin{pmatrix} 28&94&70&102\\ 44&153&112&163\end{pmatrix}$
$\begin{pmatrix} ?&?&?&?\\ ?&?&?&?\end{pmatrix} =\begin{pmatrix} 2&3\\ 3&5\end{pmatrix}^{-1} \begin{pmatrix} 28&94&70&102\\ 44&153&112&163\end{pmatrix}$
$=\frac{1}{2 \times 5 - 3 \times 3} \begin{pmatrix} 5&-3\\ -3&2\end{pmatrix} \begin{pmatrix} 28&94&70&102\\ 44&153&112&163\end{pmatrix}$
$=\begin{pmatrix} (5\times28+(-3)\times44)&(5\times94+(-3)\times153)& (5\times70+(-3)\times112)&(5\times102+(-3)\times163)\\ ((-3)\times28+2\times44)&((-3)\times94+2\times153)& ((-3)\times70+2\times112)&((-3)\times102+2\times163)\end{pmatrix}$
$=\begin{pmatrix} 8&11&14&21\\ 4&24&14&20\end{pmatrix}$
Therefore the message is "iloveyou"

2.

Combine the two matrices together, we have
$A\begin{pmatrix} 1&3\\ 2&4\end{pmatrix} =\begin{pmatrix} 1&0\\ 0&1\end{pmatrix}$
$A\begin{pmatrix} 1&3\\ 2&4\end{pmatrix} =I$
Therefore the inverse of A is
$\begin{pmatrix} 1&3\\ 2&4\end{pmatrix}$