Prove the equation

Question:

$\left|\begin{array}{ccc}a^{2}+2 a & 2 a+1 & 1 \\ 2 a+1 & a+2 & 1 \\ 3 & 3 & 1\end{array}\right|=(a-1)^{3}$

Solution:

Given, $\left|\begin{array}{ccc}a^{2}+2 a & 2 a+1 & 1 \\ 2 a+1 & a+2 & 1 \\ 3 & 3 & 1\end{array}\right|$

[Applying $R_{1} \rightarrow R_{1}-R_{2}$ and $R_{2} \rightarrow R_{2}-R_{3}$ ]

$=\left|\begin{array}{ccc}a^{2}-1 & a-1 & 0 \\ 2 a-2 & a-1 & 0 \\ 3 & 3 & 1\end{array}\right|$

Now,

[Taking $(a-1)$ common from $R_{1}$ and $R_{2}$ ]

$(a-1)^{2}\left|\begin{array}{ccc}a+1 & 1 & 0 \\ 2 & 1 & 0 \\ 3 & 3 & 1\end{array}\right|$

Finally,

[Expanding along $R_{3}$ ]

$=(a-1)^{2}[1 \cdot(a+1)-2]=(a-1)^{3}$

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