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ScanSolve AI Answer
To find the Minor and Cofactor of each element of the given determinant, we will follow these steps. The given determinant is: 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 The Minor M_ij of an element a_ij is the value of the sub-determinant obtained by deleting the i-th row and j-th column. The Cofactor C_ij of an element a_ij is given by the formula C_ij = (-1)^i+j M_ij. Step 1: Find the Minors for each element. For element a_11 = 1: M_11 = 5 & -1 \\ 1 & 2 = (5)(2) - (-1)(1) = 10 - (-1) = 11 For element a_12 = 0: M_12 = 3 & -1 \\ 0 & 2 = (3)(2) - (-1)(0) = 6 - 0 = 6 For element a_13 = 4: M_13 = 3 & 5 \\ 0 & 1 = (3)(1) - (5)(0) = 3 - 0 = 3 For element a_21 = 3: M_21 = 0 & 4 \\ 1 & 2 = (0)(2) - (4)(1) = 0 - 4 = -4 For element a_22 = 5: M_22 = 1 & 4 \\ 0 & 2 = (1)(2) - (4)(0) = 2 - 0 = 2 For element a_23 = -1: M_23 = 1 & 0 \\ 0 & 1 = (1)(1) - (0)(0) = 1 - 0 = 1 For element a_31 = 0: M_31 = 0 & 4 \\ 5 & -1 = (0)(-1) - (4)(5) = 0 - 20 = -20 For element a_32 = 1: M_32 = 1 & 4 \\ 3 & -1 = (1)(-1) - (4)(3) = -1 - 12 = -13 For element a_33 = 2: M_33 = 1 & 0 \\ 3 & 5 = (1)(5) - (0)(3) = 5 - 0 = 5 Step 2: Find the Cofactors for each element. For element a_11 = 1: C_11 = (-1)^1+1 M_11 = (-1)^2 (11) = 1 × 11 = 11 For element a_12 = 0: C_12 = (-1)^1+2 M_12 = (-1)^3 (6) = -1 × 6 = -6 For element a_13 = 4: C_13 = (-1)^1+3 M_13 = (-1)^4 (3) = 1 × 3 = 3 For element a_21 = 3: C_21 = (-1)^2+1 M_21 = (-1)^3 (-4) = -1 × (-4) = 4 For element a_22 = 5: C_22 = (-1)^2+2 M_22 = (-1)^4 (2) = 1 × 2 = 2 For element a_23 = -1: C_23 = (-1)^2+3 M_23 = (-1)^5 (1) = -1 × 1 = -1 For element a_31 = 0: C_31 = (-1)^3+1 M_31 = (-1)^4 (-20) = 1 × (-20) = -20 For element a_32 = 1: C_32 = (-1)^3+2 M_32 = (-1)^5 (-13) = -1 × (-13) = 13 For element a_33 = 2: C_33 = (-1)^3+3 M_33 = (-1)^6 (5) = 1 × 5 = 5 Final Answer: The Minors and Cofactors for each element are as follows: a_11 = 1: M_11 = 11, C_11 = 11 a_12 = 0: M_12 = 6, C_12 = -6 a_13 = 4: M_13 = 3, C_13 = 3 a_21 = 3: M_21 = -4, C_21 = 4 a_22 = 5: M_22 = 2, C_22 = 2 a_23 = -1: M_23 = 1, C_23 = -1 a_31 = 0: M_31 = -20, C_31 = -20 a_32 = 1: M_32 = -13, C_32 = 13 a_33 = 2: M_33 = 5, C_33 = 5 3 done, 2 left today. You're making progress.