Find determinant of the given matrix \( C \) \[ C=\left(\begin{array}{ccc}-5 & 4 & -5 \\ 2 & -6 & 4 \\ 6 & 2 & 4\end{array}\right) \]

To find the determinant of the matrix \( C \): \[ C=\begin{pmatrix} -5 & 4 & -5 \\ 2 & -6 & 4 \\ 6 & 2 & 4 \end{pmatrix} \] We can use the formula for the determinant of a \( 3 \times 3 \) matrix given by: \[ \text{det}(C) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where the matrix is: \[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \] For our matrix \( C \), we have: - \( a = -5, b = 4, c = -5 \) - \( d = 2, e = -6, f = 4 \) - \( g = 6, h = 2, i = 4 \) Calculating the required products: 1. \( ei - fh = (-6)(4) - (4)(2) = -24 - 8 = -32 \) 2. \( di - fg = (2)(4) - (4)(6) = 8 - 24 = -16 \) 3. \( dh - eg = (2)(2) - (-6)(6) = 4 + 36 = 40 \) Putting these values back into the determinant formula: \[ \text{det}(C) = -5(-32) - 4(-16) + (-5)(40) \] Calculating each term: - First term: \( -5(-32) = 160 \) - Second term: \( -4(-16) = 64 \) - Third term: \( -5(40) = -200 \) Now adding these results together: \[ \text{det}(C) = 160 + 64 - 200 = 24 \] Thus, the determinant of the matrix \( C \) is: \[ \boxed{24} \]