if

Question:

If $A=\left[\begin{array}{rrr}2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4\end{array}\right]$, show that $A^{2}=A$

Solution:

Here,

$A^{2}=A A$

$\Rightarrow A^{2}=\left[\begin{array}{ccc}2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4\end{array}\right]\left[\begin{array}{ccc}2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4\end{array}\right]$

$\Rightarrow A^{2}=\left[\begin{array}{ccc}4+3-5 & -6-12+15 & -10-15+20 \\ -2-4+5 & 3+16-15 & 5+20-20 \\ 2+3-4 & -3-12+12 & -5-15+16\end{array}\right]$

$\Rightarrow A^{2}=\left[\begin{array}{ccc}2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4\end{array}\right]$

$\therefore A^{2}=A$

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