To find the determinant of the given 3x3 matrix, we follow the method for computing the determinant of a 3x3 matrix.
Given the matrix
[tex]\[
\begin{bmatrix}
0 & 4 & 1 \\
5 & -3 & 0 \\
2 & 3 & 1
\end{bmatrix}
\][/tex]
We use the formula for the determinant of a 3x3 matrix:
[tex]\[
\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
where the matrix [tex]\( A \)[/tex] is:
[tex]\[
A = \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\][/tex]
For our matrix, we assign:
[tex]\[
a = 0, \quad b = 4, \quad c = 1, \quad d = 5, \quad e = -3, \quad f = 0, \quad g = 2, \quad h = 3, \quad i = 1
\][/tex]
Now substitute these values into the determinant formula:
[tex]\[
\det(A) = 0((-3 \cdot 1) - (0 \cdot 3)) - 4((5 \cdot 1) - (0 \cdot 2)) + 1((5 \cdot 3) - (-3 \cdot 2))
\][/tex]
Simplify each term separately:
First term:
[tex]\[
0((-3 \cdot 1) - (0 \cdot 3)) = 0(-3) = 0
\][/tex]
Second term:
[tex]\[
-4((5 \cdot 1) - (0 \cdot 2)) = -4(5 - 0) = -4 \cdot 5 = -20
\][/tex]
Third term:
[tex]\[
1((5 \cdot 3) - (-3 \cdot 2)) = 1(15 + 6) = 1 \cdot 21 = 21
\][/tex]
Combine them to get the determinant:
[tex]\[
\det(A) = 0 - 20 + 21 = 1
\][/tex]
Therefore, the determinant of the given matrix is [tex]\( \boxed{1} \)[/tex].
