Math 375 - 11/9

Expansion by Rows Theorem • Cofactor Matrix ○ C_kl=(−1)^(k+l) {█((n−1)×(n−1) determinant obtained@by deleting row k and column l @from the original determinant)} • Determinant and Cofactor Matrix ○ det⁡(A)=|■8(a_11&a_12&⋯&a_1n@⋮&⋮&⋮&⋮@a_k1&a_k2&…&a_kn@⋮&⋮&⋮&⋮@a_n1&a_n2&⋯&a_nn )|=a_k1 C_k1+a_k2 C_k2+…+a_kn C_kn ○ [■8(a_11&a_12&⋯&a_1n@a_21&a_22&…&a_2n@⋮&⋮&⋱&⋮@a_n1&a_n2&⋯&a_nn )] ⏟([■8(C_11&C_21&⋯&C_n1@C_12&C_22&…&C_n2@⋮&⋮&⋱&⋮@C_1n&C_2n&⋯&C_nn )] )┬(adjugate matrix of A: adj(A))=det⁡A⋅[■(1&&&@&1&&@&&⋱&@&&&1)] • Expansion by Rows ○ |■8(a_11&a_12&⋯&a_1n@a_21&a_22&…&a_2n@⋮&⋮&⋱&⋮@a_n1&a_n2&⋯&a_nn )|=a_11 C_11+a_12 C_12+…+a_1n C_1n ○ |■8(x_1&x_2&⋯&x_n@a_21&a_22&…&a_2n@⋮&⋮&⋱&⋮@a_n1&a_n2&⋯&a_nn )|=x_1 C_11+x_2 C_12+…+x_n C_1n • Calculating A⋅adj(A) ○ Expanding A⋅adj(A) § [■8(a_11&a_12&⋯&a_1n@a_21&a_22&…&a_2n@⋮&⋮&⋱&⋮@a_n1&a_n2&⋯&a_nn )][■8(C_11&C_21&⋯&C_n1@C_12&C_22&…&C_n2@⋮&⋮&⋱&⋮@C_1n&C_2n&⋯&C_nn )] § =[■8(∑_(k=1)^n▒〖a_1k C_1k 〗&∑_(k=1)^n▒〖a_1k C_2k 〗&⋯&∑_(k=1)^n▒〖a_1k C_nk 〗@∑_(k=1)^n▒〖a_2k C_1k 〗&∑_(k=1)^n▒〖a_2k C_2k 〗&…&∑_(k=1)^n▒〖a_2k C_nk 〗@⋮&⋮&⋱&⋮@∑_(k=1)^n▒〖a_nk C_1k 〗&∑_(k=1)^n▒〖a_nk C_2k 〗&⋯&∑_(k=1)^n▒〖a_nk C_nk 〗)] ○ Where § ∑_(k=1)^n▒〖a_1k C_1k 〗=|■8(a_11&a_12&⋯&a_1n@a_21&a_22&…&a_2n@⋮&⋮&⋱&⋮@a_n1&a_n2&⋯&a_nn )|=det⁡A § ∑_(k=1)^n▒〖a_1k C_2k 〗=|■8(a_21&a_22&⋯&a_2n@a_21&a_22&…&a_2n@⋮&⋮&⋱&⋮@a_n1&a_n2&⋯&a_nn )|=0 § ⋮ ○ Conclusion § A⋅adj(A)=[■(det⁡A&&&@&det⁡A&&@&&⋱&@&&&det⁡A )]=det⁡A [■(1&&&@&1&&@&&⋱&@&&&1)] • Theorem ○ det⁡〖(A)≠0〗⟺A is invertible and A^(−1)=1/det⁡A ⋅adj(A) ○ det⁡(A)=0⟺A is not invertible • Example ○ Let A=[■8(a&b@c&d)] ○ Cofactor Matrix § C=[■8(C_11&C_12@C_21&C_22 )]=[■8(d&−c@−b&a)] ○ Adjugate Matrix § adj(A)=C^T=[■8(d&−b@−c&a)] ○ Determinant § det⁡A=|■8(a&b@c&d)|=ad−bc ○ Inverse Matrix § A^{(−1)=[■8(a&b@c&d)]}(−1)=1/det⁡A ⋅adj(A)=1/(ad−bc) [■8(d&−b@−c&a)] Cramer