Determinant of $n\times n$ matrix How can I calculate the determinant of the following $n\times n$ matrix, where $n$ is a multiple of $3$ ?
$$\begin{pmatrix}
0 & 0 & 1 & & & & & & &\\
& & & 0 & 0 & 1 & & & &\\
& & & & & &\ddots\\
& & & & & & & 0 & 0 & 1\\
0 & 1 & 0 & & & & & & &\\
& & & 0 & 1 & 0 & & & &\\
& & & & & &\ddots\\
& & & & & & & 0 & 1 & 0\\
1 & 0 & 0 & & & & & & &\\
& & & 1 & 0 & 0 & & & &\\
& & & & & &\ddots\\
& & & & & & & 1 & 0 & 0
\end{pmatrix}$$
 A: You can use induction on $m$, using $n = 3m, \quad m\in \mathbb{N}$. 
First, compute the determinants for base cases $m = 1$ where $n = 3\cdot 1 =3$, and, say, $m=2$, where $n = 3 \cdot 2 = 6$. (It would also be insightful to see what happens for a couple more "smallish" $m$.)
You'll find by using elementary row operations, you can obtain the identity matrix for any $n = 3m$. The only elementary row operation you'll need that changes the value of the original determinant is "exchange row i with row j", so you need only record the number of times "row exchangement" is required : 


*

*When an odd number of exchanges is needed (e.g., n = 3), the determinant will be $-1$. 

*When an even number of exchanges is required, then the determinant will be 1.


Through exploration, you'll note a pattern
After computing the determinants for $n = 3m$ for "small-ish" values of $m$, you should be able to establish a pattern and determine an expression in terms of $m$ that represents the required number of exchanges for any given $n = 3\cdot m$. 
Use this expression to propose an inductive hypothesis, and using the inductive hypothesis, see if you can show that for $n = 3\cdot (m + 1)$, the inductive hypothesis is satisfied.
