If A= [ cc 2 & -1 -1 & 2 ], then show that A^2-4 A+3l=0. - Vidyadip

Question

If $A=\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]$, then show that $A^2-4 A+3l=0$.

✓

Answer

$
\begin{aligned}
A ^2 & = A \cdot A =\left[\begin{array}{rr}
2 & -1 \\
-1 & 2
\end{array}\right]\left[\begin{array}{rr}
2 & -1 \\
-1 & 2
\end{array}\right] \\
& =\left[\begin{array}{rr}
4+1 & -2-2 \\
-2-2 & 1+4
\end{array}\right]=\left[\begin{array}{rr}
5 & -4 \\
-4 & 5
\end{array}\right] \\
\therefore & A ^2-4 A +3 I \\
& =\left[\begin{array}{rr}
5 & -4 \\
-4 & 5
\end{array}\right]-4\left[\begin{array}{rr}
2 & -1 \\
-1 & 2
\end{array}\right]+3\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]
\end{aligned}
$
$
\begin{aligned}
& =\left[\begin{array}{rr}
5 & -4 \\
-4 & 5
\end{array}\right]-\left[\begin{array}{rr}
8 & -4 \\
-4 & 8
\end{array}\right]+\left[\begin{array}{ll}
3 & 0 \\
0 & 3
\end{array}\right] \\
& =\left[\begin{array}{rr}
5-8+3 & -4-(-4)+0 \\
-4-(-4)+0 & 5-8+3
\end{array}\right]=\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right] \\
\therefore & A^2-4 A+3 I=0 .
\end{aligned}
$

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