High School Mathematics Extensions/Matrices/Problem Set/Solutions

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Matrices Problem Set[edit]

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 Therefore this exercise solutions section may be out of date and appear disorganised.

If you have a question please leave a comment in the "discussion section" or contact the author or any of the major contributors.


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}