Question
If $A=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]$, find $A 2$ and $A 3$. Also state that which of these is equal to $A$
✓
Answer
$
\begin{aligned}
& A=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right] \\
& A^2=A \times A=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right] \\
& =\left[\begin{array}{ll}
1+0 & 0+0 \\
0+0 & 0+1
\end{array}\right] \\
& =\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] \\
& A^3=A^2+A=\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right] \times\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1+0 & 0+0 \\
0+0 & 0-1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]
\end{aligned}
$
From above it is clear that $A^3=A$.
\begin{aligned}
& A=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right] \\
& A^2=A \times A=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right] \\
& =\left[\begin{array}{ll}
1+0 & 0+0 \\
0+0 & 0+1
\end{array}\right] \\
& =\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] \\
& A^3=A^2+A=\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right] \times\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1+0 & 0+0 \\
0+0 & 0-1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]
\end{aligned}
$
From above it is clear that $A^3=A$.
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