Given the matrix: \( A=\left[\begin{array}{lll}8 & 5 & 3 \\ 4 & 7 & 2 \\ 6 & 1 & 9\end{array}\right] \) Find the cofactor of \( A \)

To find the cofactor matrix of A, we compute the cofactor for each element a₍ᵢⱼ₎ using the formula:   C₍ᵢⱼ₎ = (–1)^(i+j) · M₍ᵢⱼ₎ where M₍ᵢⱼ₎ is the determinant of the submatrix formed by deleting the i‑th row and j‑th column from A. The matrix A is:   A = [ [8, 5, 3],       [4, 7, 2],       [6, 1, 9] ] Step 1. Compute the minors and cofactors. 1. For element a₁₁ = 8 (i = 1, j = 1):   Delete row 1 and column 1:      [ [7, 2],        [1, 9] ]   Determinant = (7)(9) – (2)(1) = 63 – 2 = 61.   Sign = (–1)^(1+1) = (–1)^2 = 1.   C₁₁ = 1 · 61 = 61. 2. For element a₁₂ = 5 (i = 1, j = 2):   Delete row 1 and column 2:      [ [4, 2],        [6, 9] ]   Determinant = (4)(9) – (2)(6) = 36 – 12 = 24.   Sign = (–1)^(1+2) = (–1)^3 = –1.   C₁₂ = –1 · 24 = –24. 3. For element a₁₃ = 3 (i = 1, j = 3):   Delete row 1 and column 3:      [ [4, 7],        [6, 1] ]   Determinant = (4)(1) – (7)(6) = 4 – 42 = –38.   Sign = (–1)^(1+3) = (–1)^4 = 1.   C₁₃ = 1 · (–38) = –38. 4. For element a₂₁ = 4 (i = 2, j = 1):   Delete row 2 and column 1:      [ [5, 3],        [1, 9] ]   Determinant = (5)(9) – (3)(1) = 45 – 3 = 42.   Sign = (–1)^(2+1) = (–1)^3 = –1.   C₂₁ = –1 · 42 = –42. 5. For element a₂₂ = 7 (i = 2, j = 2):   Delete row 2 and column 2:      [ [8, 3],        [6, 9] ]   Determinant = (8)(9) – (3)(6) = 72 – 18 = 54.   Sign = (–1)^(2+2) = (–1)^4 = 1.   C₂₂ = 1 · 54 = 54. 6. For element a₂₃ = 2 (i = 2, j = 3):   Delete row 2 and column 3:      [ [8, 5],        [6, 1] ]   Determinant = (8)(1) – (5)(6) = 8 – 30 = –22.   Sign = (–1)^(2+3) = (–1)^5 = –1.   C₂₃ = –1 · (–22) = 22. 7. For element a₃₁ = 6 (i = 3, j = 1):   Delete row 3 and column 1:      [ [5, 3],        [7, 2] ]   Determinant = (5)(2) – (3)(7) = 10 – 21 = –11.   Sign = (–1)^(3+1) = (–1)^4 = 1.   C₃₁ = 1 · (–11) = –11. 8. For element a₃₂ = 1 (i = 3, j = 2):   Delete row 3 and column 2:      [ [8, 3],        [4, 2] ]   Determinant = (8)(2) – (3)(4) = 16 – 12 = 4.   Sign = (–1)^(3+2) = (–1)^5 = –1.   C₃₂ = –1 · 4 = –4. 9. For element a₃₃ = 9 (i = 3, j = 3):   Delete row 3 and column 3:      [ [8, 5],        [4, 7] ]   Determinant = (8)(7) – (5)(4) = 56 – 20 = 36.   Sign = (–1)^(3+3) = (–1)^6 = 1.   C₃₃ = 1 · 36 = 36. Step 2. Write the cofactor matrix using the computed cofactors.   Cofactor matrix, Cof(A), is:   [ [C₁₁, C₁₂, C₁₃],     [C₂₁, C₂₂, C₂₃],     [C₃₁, C₃₂, C₃₃] ]   = [ [61, –24, –38],      [–42, 54, 22],      [–11, –4, 36] ] Thus, the cofactor matrix of A is:   [ [61 -24 -38],    [–42 54 22],    [–11 -4 36] ].