Compute
\[\begin{vmatrix} 1 & -3 & 3 \\ 0 & 5 & -1 \\ 4 & -2 & 1 \end{vmatrix}.\]
We can expand the determinant as follows:
\begin{align*}
\begin{vmatrix} 1 & -3 & 3 \\ 0 & 5 & -1 \\ 4 & -2 & 1 \end{vmatrix} &= \begin{vmatrix} 5 & -1 \\ -2 & 1 \end{vmatrix} - (-3) \begin{vmatrix} 0 & -1 \\ 4 & 1 \end{vmatrix} + 3 \begin{vmatrix} 0 & 5 \\ 4 & -2 \end{vmatrix} \\
&= ((5)(1) - (-1)(-2)) + 3 ((0)(1) - (-1)(4)) + 3 ((0)(-2) - (5)(4)) \\
&= \boxed{-45}.
\end{align*}We can also expand along the first column, to take advantage of the 0 in the first column, to get
\begin{align*}
\begin{vmatrix} 1 & -3 & 3 \\ 0 & 5 & -1 \\ 4 & -2 & 1 \end{vmatrix} &= \begin{vmatrix} 5 & -1 \\ -2 & 1 \end{vmatrix} + 4 \begin{vmatrix} -3 & 3 \\ 5 & -1 \end{vmatrix} \\
&= ((5)(1) - (-1)(-2)) + 4((-3)(-1) - (3)(5)) \\
&= \boxed{-45}.
\end{align*}
-45
Level 3
Precalculus
Problem: Compute
\[\begin{vmatrix} 1 & -3 & 3 \\ 0 & 5 & -1 \\ 4 & -2 & 1 \end{vmatrix}.\]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To compute the determinant of the given 3x3 matrix, we apply the formula for the determinant of a 3x3 matrix, which is:

det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

where the matrix is represented as:

\[\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}.\]

For the matrix 

\[\begin{pmatrix} 1 & -3 & 3 \\ 0 & 5 & -1 \\ 4 & -2 & 1 \end{pmatrix},\]

we identify the elements as follows:
a = 1, b = -3, c = 3,
d = 0, e = 5, f = -1,
g = 4, h = -2, i = 1.

Now, we substitute these values into the determinant formula:

det(A) = 1(5*1 - (-1)(-2)) - (-3)(0*1 - (-1)(4)) + 3(0*(-2) - 5*4).

Calculating each term step-by-step:

1. For the first term: 
5*1 = 5 and (-1)(-2) = 2, so 5 - 2 = 3. Thus, the first term is 1 * 3 = 3.

2. For the second term:
0*1 = 0 and (-1)(4) = -4, so 0 - (-4) = 4. Thus, the second term is -(-3) * 4 = 3 * 4 = 12.

3. For the third term:
0*(-2) = 0 and 5*4 = 20, so 0 - 20 = -20. Thus, the third term is 3 * (-20) = -60.

Now, we combine all the terms:

det(A) = 3 + 12 - 60 = 15 - 60 = -45.

Therefore, the final answer is -45.<|END_OF_LATENT|> The final answer is $-45$.