Given the matrix \( A=\left[\begin{array}{ccc}4 & 1 & 2 \\ 1 & -6 & 3 \\ -2 & 5 & 9\end{array}\right] \) and \( B=\left[\begin{array}{ccc}2 & -3 & 4 \\ 1 & 5 & 6 \\ 8 & 2 & -7\end{array}\right] \) find \( |A+B| \)

To find the determinant \( |A+B| \), we first need to compute the matrix \( A+B \). Calculating \( A + B \): \[ A + B = \begin{bmatrix} 4 & 1 & 2 \\ 1 & -6 & 3 \\ -2 & 5 & 9 \end{bmatrix} + \begin{bmatrix} 2 & -3 & 4 \\ 1 & 5 & 6 \\ 8 & 2 & -7 \end{bmatrix} \] Adding corresponding elements: \[ A + B = \begin{bmatrix} 4+2 & 1+(-3) & 2+4 \\ 1+1 & -6+5 & 3+6 \\ -2+8 & 5+2 & 9+(-7) \end{bmatrix} = \begin{bmatrix} 6 & -2 & 6 \\ 2 & -1 & 9 \\ 6 & 7 & 2 \end{bmatrix} \] Now we need to calculate the determinant \( |A+B| = | \begin{bmatrix} 6 & -2 & 6 \\ 2 & -1 & 9 \\ 6 & 7 & 2 \end{bmatrix} | \). Using the formula for the determinant of a 3x3 matrix \( |M| = a(ei-fh) - b(di-fg) + c(dh-eg) \): \[ \text{Let } M = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \] Where \( a = 6, b = -2, c = 6, d = 2, e = -1, f = 9, g = 6, h = 7, i = 2 \). Now, substituting the values into the determinant formula: \[ |M| = 6((-1)(2) - (9)(7)) - (-2)(2(2) - (9)(6)) + 6(2(7) - (-1)(6)) \] Calculating each part step-by-step: 1. First term: \( 6((-2) - 63) = 6(-65) = -390 \) 2. Second term: \( -2(4 - 54) = -2(-50) = 100 \) 3. Third term: \( 6(14 + 6) = 6(20) = 120 \) Combining these: \[ |M| = -390 + 100 + 120 = -170 \] Therefore, the determinant \( |A+B| \) is: \[ \boxed{-170} \]