The problem is to find the determinant of the given matrix: $ \begin{bmatrix} 1 & 3 & 2 \\ 0 & 4 & 3 \\ 2 & -5 & 4 \end{bmatrix} $

1. Problem Description

The problem is to find the determinant of the given matrix:

$\begin{bmatrix} 1 & 3 & 2 \\ 0 & 4 & 3 \\ 2 & -5 & 4 \end{bmatrix}$

2. Solution Steps

The determinant of a 3x3 matrix

$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$

is given by:

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

In our case, we have:

$a = 1, b = 3, c = 2$

$d = 0, e = 4, f = 3$

$g = 2, h = -5, i = 4$

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

$det = 1(16 + 15) - 3(0 - 6) + 2(0 - 8)$

$det = 1(31) - 3(-6) + 2(-8)$

$det = 31 + 18 - 16$

$det = 49 - 16$

$det = 33$

3. Final Answer

The determinant of the matrix is

3

3. So the answer is C.