1. We are asked to find the determinant of the matrix: $$\begin{bmatrix} 5 & 8 & 14 \\ 6 & 12 & 13 \\ 8 & 5 & 6 \end{bmatrix}$$ 2. The formula for the determinant of a 3x3 matrix $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ is: $$\det = a(ei - fh) - b(di - fg) + c(dh - eg)$$ 3. Substitute the values from the given matrix: $$a=5, b=8, c=14, d=6, e=12, f=13, g=8, h=5, i=6$$ 4. Calculate each part: - $ei - fh = 12 \times 6 - 13 \times 5 = 72 - 65 = 7$ - $di - fg = 6 \times 6 - 13 \times 8 = 36 - 104 = -68$ - $dh - eg = 6 \times 5 - 12 \times 8 = 30 - 96 = -66$ 5. Now compute the determinant: $$5 \times 7 - 8 \times (-68) + 14 \times (-66) = 35 + 544 - 924 = (35 + 544) - 924 = 579 - 924 = -345$$ 6. Therefore, the determinant of the matrix is: $$\boxed{-345}$$