High School Mathematics Extensions/Matrices/Problem Set/Solutions

Matrices Problem Set

1.

$\left(\begin{array}{cc}2& 3\\ 3& 5\end{array}\right)\left(\begin{array}{cccc}?& ?& ?& ?\\ ?& ?& ?& ?\end{array}\right)=\left(\begin{array}{cccc}28& 94& 70& 102\\ 44& 153& 112& 163\end{array}\right)$

${\left(\begin{array}{cc}2& 3\\ 3& 5\end{array}\right)}^{-1}\left(\begin{array}{cc}2& 3\\ 3& 5\end{array}\right)\left(\begin{array}{cccc}?& ?& ?& ?\\ ?& ?& ?& ?\end{array}\right)={\left(\begin{array}{cc}2& 3\\ 3& 5\end{array}\right)}^{-1}\left(\begin{array}{cccc}28& 94& 70& 102\\ 44& 153& 112& 163\end{array}\right)$

$\left(\begin{array}{cccc}?& ?& ?& ?\\ ?& ?& ?& ?\end{array}\right)={\left(\begin{array}{cc}2& 3\\ 3& 5\end{array}\right)}^{-1}\left(\begin{array}{cccc}28& 94& 70& 102\\ 44& 153& 112& 163\end{array}\right)$

$=\frac{1}{2\times 5-3\times 3}\left(\begin{array}{cc}5& -3\\ -3& 2\end{array}\right)\left(\begin{array}{cccc}28& 94& 70& 102\\ 44& 153& 112& 163\end{array}\right)$

$=\left(\begin{array}{cccc}(5\times 28+(-3)\times 44)& (5\times 94+(-3)\times 153)& (5\times 70+(-3)\times 112)& (5\times 102+(-3)\times 163)\\ ((-3)\times 28+2\times 44)& ((-3)\times 94+2\times 153)& ((-3)\times 70+2\times 112)& ((-3)\times 102+2\times 163)\end{array}\right)$

$=\left(\begin{array}{cccc}8& 11& 14& 21\\ 4& 24& 14& 20\end{array}\right)$

Therefore the message is "iloveyou"

2.

Combine the two matrices together, we have

$A\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

$A\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)=I$

Therefore the inverse of A is

$\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)$