(5) given that A = [[5, 0, 8], [12, 7, 6], [6, 4, 4]]. Evaluate A^-1

To evaluate A^-1 for the given matrix A, we will use the formula A^-1 = (1)/((A)) adj(A), where (A) is the determinant of A and adj(A) is the adjoint of A. The given matrix is: A = 5 & 0 & 8 \\ 12 & 7 & 6 \\ 6 & 4 & 4 Step 1: Calculate the determinant of A. (A) = 5 7 & 6 \\ 4 & 4 - 0 12 & 6 \\ 6 & 4 + 8 12 & 7 \\ 6 & 4 (A) = 5(7 × 4 - 6 × 4) - 0(12 × 4 - 6 × 6) + 8(12 × 4 - 7 × 6) (A) = 5(28 - 24) - 0 + 8(48 - 42) (A) = 5(4) + 8(6) (A) = 20 + 48 (A) = 68 Step 2: Calculate the cofactor matrix C. The cofactor C_ij is given by (-1)^i+j M_ij, where M_ij is the minor of the element a_ij. C_11 = 7 & 6 \\ 4 & 4 = 28 - 24 = 4 C_12 = - 12 & 6 \\ 6 & 4 = -(48 - 36) = -12 C_13 = 12 & 7 \\ 6 & 4 = 48 - 42 = 6 C_21 = - 0 & 8 \\ 4 & 4 = -(0 - 32) = 32 C_22 = 5 & 8 \\ 6 & 4 = 20 - 48 = -28 C_23 = - 5 & 0 \\ 6 & 4 = -(20 - 0) = -20 C_31 = 0 & 8 \\ 7 & 6 = 0 - 56 = -56 C_32 = - 5 & 8 \\ 12 & 6 = -(30 - 96) = -(-66) = 66 C_33 = 5 & 0 \\ 12 & 7 = 35 - 0 = 35 The cofactor matrix is: C = 4 & -12 & 6 \\ 32 & -28 & -20 \\ -56 & 66 & 35 Step 3: Calculate the adjoint of A. The adjoint of A is the transpose of the cofactor matrix C. adj(A) = C^T = 4 & 32 & -56 \\ -12 & -28 & 66 \\ 6 & -20 & 35 Step 4: Calculate the inverse of A. A^-1 = (1)/((A)) adj(A) = (1)/(68) 4 & 32 & -56 \\ -12 & -28 & 66 \\ 6 & -20 & 35 Distribute the scalar (1)/(68) into the matrix and simplify the fractions: A^-1 = (4)/(68) & (32)/(68) & (-56)/(68) \\ (-12)/(68) & (-28)/(68) & (66)/(68) \\ (6)/(68) & (-20)/(68) & (35)/(68) = (1)/(17) & (8)/(17) & (-14)/(17) \\ (-3)/(17) & (-7)/(17) & (33)/(34) \\ (3)/(34) & (-5)/(17) & (35)/(68) The inverse of matrix A is: A^-1 = (1)/(17) & (8)/(17) & (-14)/(17) \\ (-3)/(17) & (-7)/(17) & (33)/(34) \\ (3)/(34) & (-5)/(17) & (35)/(68) What's next? Send 'em! 📸