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To find the minors and cofactors of the elements of the given determinant: 1 & 2 & -3 \\ -2 & 0 & 4 \\ 5 & -1 & 3 Let the matrix be A = a_11 & a_12 & a_13 \\ a_21 & a_22 & a_23 \\ a_31 & a_32 & a_33 . The minor M_ij is the determinant of the submatrix formed by deleting row i and column j. The cofactor C_ij is given by C_ij = (-1)^i+j M_ij. For element a_11 = 1: Step 1: Find the minor M_11. M_11 = 0 & 4 \\ -1 & 3 = (0)(3) - (4)(-1) = 0 - (-4) = 4 Step 2: Find the cofactor C_11. C_11 = (-1)^1+1 M_11 = (1)(4) = 4 For element a_12 = 2: Step 1: Find the minor M_12. M_12 = -2 & 4 \\ 5 & 3 = (-2)(3) - (4)(5) = -6 - 20 = -26 Step 2: Find the cofactor C_12. C_12 = (-1)^1+2 M_12 = (-1)(-26) = 26 For element a_13 = -3: Step 1: Find the minor M_13. M_13 = -2 & 0 \\ 5 & -1 = (-2)(-1) - (0)(5) = 2 - 0 = 2 Step 2: Find the cofactor C_13. C_13 = (-1)^1+3 M_13 = (1)(2) = 2 For element a_21 = -2: Step 1: Find the minor M_21. M_21 = 2 & -3 \\ -1 & 3 = (2)(3) - (-3)(-1) = 6 - 3 = 3 Step 2: Find the cofactor C_21. C_21 = (-1)^2+1 M_21 = (-1)(3) = -3 For element a_22 = 0: Step 1: Find the minor M_22. M_22 = 1 & -3 \\ 5 & 3 = (1)(3) - (-3)(5) = 3 - (-15) = 18 Step 2: Find the cofactor C_22. C_22 = (-1)^2+2 M_22 = (1)(18) = 18 For element a_23 = 4: Step 1: Find the minor M_23. M_23 = 1 & 2 \\ 5 & -1 = (1)(-1) - (2)(5) = -1 - 10 = -11 Step 2: Find the cofactor C_23. C_23 = (-1)^2+3 M_23 = (-1)(-11) = 11 For element a_31 = 5: Step 1: Find the minor M_31. M_31 = 2 & -3 \\ 0 & 4 = (2)(4) - (-3)(0) = 8 - 0 = 8 Step 2: Find the cofactor C_31. C_31 = (-1)^3+1 M_31 = (1)(8) = 8 For element a_32 = -1: Step 1: Find the minor M_32. M_32 = 1 & -3 \\ -2 & 4 = (1)(4) - (-3)(-2) = 4 - 6 = -2 Step 2: Find the cofactor C_32. C_32 = (-1)^3+2 M_32 = (-1)(-2) = 2 For element a_33 = 3: Step 1: Find the minor M_33. M_33 = 1 & 2 \\ -2 & 0 = (1)(0) - (2)(-2) = 0 - (-4) = 4 Step 2: Find the cofactor C_33. C_33 = (-1)^3+3 M_33 = (1)(4) = 4 The minors and cofactors are: M_11 = 4, C_11 = 4 M_12 = -26, C_12 = 26 M_13 = 2, C_13 = 2 M_21 = 3, C_21 = -3 M_22 = 18, C_22 = 18 M_23 = -11, C_23 = 11 M_31 = 8, C_31 = 8 M_32 = -2, C_32 = 2 M_33 = 4, C_33 = 4 The matrix of minors is: 4 & -26 & 2 \\ 3 & 18 & -11 \\ 8 & -2 & 4 The matrix of cofactors is: 4 & 26 & 2 \\ -3 & 18 & 11 \\ 8 & 2 & 4