High School Mathematics Extensions/Matrices/Problem Set/Solutions

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Matrices Problem Set

At the moment, the main focus is on authoring the main content of each chapter. Therefore this exercise solutions section may be out of date and appear disorganised.

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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

$\begin{pmatrix} 1&3\\ 2&4\end{pmatrix}$