<s>[INST] Incorrectly solving the determinant of a matrix Compute $det(B^4)$, where $B = 
    \begin{bmatrix}
     1       & 0 & 1 \\
     1       & 1 & 2 \\
     1       & 2 & 1 
    \end{bmatrix}
$ 
I created
$C=\begin{bmatrix}
     1       & 2 \\
     1       & 1 \\
    \end{bmatrix}
$ and 
$
 D= \begin{bmatrix}
     1       & 1 \\
     1       & 2 \\
    \end{bmatrix}
$
$det(B) = -det(C) -2det(D) = -(1-2) - 2(2-1) = -(-1)-2(1)=1-2=1, 1^4=1$
However, the correct answer is 16. I'm confused on where I made my wrong turn 
 [/INST] We have
$$ 
\det(B) = 1 \times \det(E) + 1 \times \det(D) = (1 - 4) + (2 - 1) = -2
$$
where $ E $ is 
$$
\begin{bmatrix}
1 & 2 \\ 2 & 1 
\end{bmatrix}
$$
Then 
$$
\det(B^4) = \det(B)^4 = (-2)^4 = 16
$$
</s>