A better way to evaluate a certain determinant

Create two zeroes in the first row ($C_2 \to C_2-\color{red}{4}C_1$ and $C_3 \to C_3-\color{blue}{9}C_1$) and expand: $$\begin{vmatrix} 1 & \color{red}{4} & \color{blue}{9} \\ 4 & 9 & 16 \\ 9 & 16 & 25 \\ \end{vmatrix}= \begin{vmatrix} 1 & 0 & 0 \\ 4 & 9-16 & 16-36 \\ 9 & 16-36 & 25-81 \\ \end{vmatrix} = \begin{vmatrix} -7 & -20 \\ -20 & -56 \\ \end{vmatrix} = 7 \cdot 56 - 20^2 = -8$$


Using the rule of Sarrus, the computation is really not too long, and we get in general for all $n\ge 1$, $$ \det \begin{pmatrix} n^2 & (n+1)^2 & (n+2)^2\cr (n+1)^2& (n+2)^2 & (n+3)^2\cr (n+2)^2& (n+3)^2 & (n+4)^2\end{pmatrix}=-8. $$


The direct formula for $3$ by $3$ determinants isn't so bad

$$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{vmatrix}=aei+bfg+cdh-ceg-bdi-afh$$

so

$$\begin{vmatrix} 1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25 \\ \end{vmatrix}=225+576+576-729-400-256=-8.$$

Row operations and other similar tricks tend to speed things up only when the matrix is $4$ by $4$ or larger.

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Determinant

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