<s>[INST] How to find the determinant of this $3 \times 3$ Hankel matrix? Today, at my linear algebra exam, there was this question that I couldn't solve.


Prove that
$$\det \begin{bmatrix} 
n^{2} & (n+1)^{2} &(n+2)^{2} \\ 
(n+1)^{2} &(n+2)^{2}  & (n+3)^{2}\\ 
(n+2)^{2} & (n+3)^{2} & (n+4)^{2}
\end{bmatrix} = -8$$


Clearly, calculating the determinant, with the matrix as it is, wasn't the right way. The calculations went on and on. But I couldn't think of any other way to solve it.
Is there any way to simplify $A$, so as to calculate the determinant?
 [/INST] Recall that $a^2-b^2=(a+b)(a-b)$. Subtracting $\operatorname{Row}_1$ from $\operatorname{Row}_2$ and from $\operatorname{Row}_3$ gives 
$$
\begin{bmatrix}
n^2 & (n+1)^2 & (n+2)^2 \\
2n+1 & 2n+3 & 2n+5 \\
4n+4 & 4n+8 & 4n+12
\end{bmatrix}
$$
Then subtracting $2\cdot\operatorname{Row}_2$ from $\operatorname{Row}_3$ gives
$$
\begin{bmatrix}
n^2 & (n+1)^2 & (n+2)^2 \\
2n+1 & 2n+3 & 2n+5 \\
2 & 2 & 2
\end{bmatrix}
$$
Now, subtracting $\operatorname{Col}_1$ from $\operatorname{Col}_2$ and $\operatorname{Col}_3$ gives
$$
\begin{bmatrix}
n^2 & 2n+1 & 4n+4 \\
2n+1 & 2 & 4 \\
2 & 0 & 0
\end{bmatrix}
$$
Finally, subtracting $2\cdot\operatorname{Col}_2$ from $\operatorname{Col}_3$ gives 
$$
\begin{bmatrix}
n^2 & 2n+1 & 2 \\
2n+1 & 2 & 0 \\
2 & 0 & 0
\end{bmatrix}
$$
Expanding the determinant about $\operatorname{Row}_3$ gives
$$
\det A
=
2\cdot\det
\begin{bmatrix}
2n+1 & 2\\
2 & 0
\end{bmatrix}
=2\cdot(-4)=-8
$$
as advertised.
</s>