To find the determinant of the matrix \(P = \begin{pmatrix} 1 & 3 & 2 \\ 4 & 5 & -1 \\ -3 & 2 & 0 \end{pmatrix}\) We use the formula: \(\text{det}(P) = a(ei - fh) - b(di - fg) + c(dh - eg)\) Substituting the values: - \(a = 1\), \(b = 3\), \(c = 2\) - \(d = 4\), \(e = 5\), \(f = -1\) - \(g = -3\), \(h = 2\), \(i = 0\) Calculating: \(ei - fh = 5 \cdot 0 - (-1) \cdot 2 = 2,\) \(di - fg = 4 \cdot 0 - (-1) \cdot -3 = -3,\) \(dh - eg = 4 \cdot 2 - 5 \cdot -3 = 8 + 15 = 23.\) Thus, \(\text{det}(P) = 1 \cdot 2 - 3 \cdot (-3) + 2 \cdot 23 = 2 + 9 + 46 = 57\) The determinant of matrix \( P \) is 57