16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-19A =⎡⎢⎣⎤a 11 a 12 a 13⎥a 21 a 22 a 23 ⎦a 31 a 32 a 33Let u i denote the i−th column of A −1 .Then,⎡∣ a 22 a ∣∣∣∣ ⎤23∣ a 32 a 33 ∣ 1au 1 =− 21 a ∣∣∣∣23, udet(A)∣a 31 a 33 2 =⎢∣ ⎣a 21 a ∣∣∣∣ ⎥22 ⎦∣ a 31 a 321det(A)⎡⎢⎣−∣∣−∣∣a 12 a ∣∣∣∣ ⎤13a 32 a 33 ∣ a 11 a ∣∣∣∣13a 31 a 33 ∣ a 11 a ∣∣∣∣ ⎥12 ⎦a 31 a 32u 3 =1det(A)⎡⎢⎣∣−∣∣∣a 12 a ∣∣∣∣ ⎤13a 22 a 23 ∣ a 11 a ∣∣∣∣13a 21 a 23 ∣ a 11 a ∣∣∣∣ ⎥12 ⎦a 21 a 22In practice, this easy to compute. After computing the determinant ofA, it simply involves computing 9 2 × 2 determinants.5.2 The inverse of an n × n matrix ALet A be an n × n matrix.For a given pair (i, j) of indexes (i.e., 1 ≤ i ≤ n, 1 ≤ j ≤ n), define then × n matrix C = Adj(A) byC ij = (−1) i+j det(A(j | i))