Identities, Inverses and Determinants for 2×2 Matrices
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Given \(A = \left[ {\begin{array}{*{20}{c}}2&3\\1&4\end{array}} \right]\) find \({A^{ - 1}}.\)
\begin{align}
&\text { If } A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \text { then } A^{-1}=\frac{1}{\operatorname{det}(A)}\left[\begin{array}{cc}
d & -b \\
-c & a
\end{array}\right] \\
&\operatorname{det}(A)=\left|\begin{array}{ll}
a & b \\
c & d
\end{array}\right|=a d-b c . \\
&\begin{aligned}
\text { If } A&=\left[\begin{array}{ll}
2 & 3 \\
1 & 4
\end{array}\right] \\
A^{-1}&=\frac{1}{(8-3)}\left[\begin{array}{cc}
4 & -3 \\
-1 & 2
\end{array}\right] \\
&=\frac{1}{5}\left[\begin{array}{cc}
4 & -3 \\
-1 & 2
\end{array}\right]\\
&=\left[\begin{array}{cc}
0.8 & -0.6 \\
-0.2 & 0.4
\end{array}\right]
\end{aligned}
\end{align}
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