Identity and Inverse Matrices
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Which of the following is the identity matrix?
\(\left[ {\begin{array}{*{20}{c}}1&1\\0&0\end{array}} \right]\)
\(\left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\)
\(\left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]\)
\(\left[ {\begin{array}{*{20}{c}}0&0\\1&1\end{array}} \right]\)
\begin{align}
&\text{Test } \left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\\
&\begin{aligned}
\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right] \times\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] &=\left[\begin{array}{ll}
1 \times 1+2 \times 0 & 1 \times 0+2 \times 1 \\
3 \times 1+ 4 \times 0 & 3 \times 0+4 \times 1
\end{array}\right] \\
&=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]
\end{aligned}\\
&\therefore\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \text{ is the identity matrix}
\end{align}
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