MATH125 Lecture Notes - Lecture 19: Coefficient Matrix, Row And Column Spaces, Linear Combination

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: George Peschke

Document Summary

Then the system has a unique solution, ans this solution has xj = det[a_1_1 a_1_j-1 c1 a_1_j+1 a_1_n] [a_n_1 a_n_j-1 cn a_n_j+1 a_n_n] det[a_1_1 a_1_j-1 a_1_j a_1_j+1 a_1_n] Numerator matrix: replace j-th column of a by [c1 cn]^t. = det [a_1 a_j-1 c a_j+1 a_n] = det [a_1 a_j-1 (x_1 a_1 + + x_j-1 a_j-1 + x_j a_j + x_j+1. + det [a_1 a_j-1 x_j-1 a_j-1 a_j+1 a_n] + det [a_1 a_j-1 x_j a_j a_j+1 a_n] + det [a_1 a_j+1 x_j+1 a_j+1 a_j+1 a_n] + + det [a_1 a_j-1 x_n a_n a_j+1 a_n] 0 + + 0 + det [a_1 a_j x_j a_j a_j a_n] + 0 + + 0. Means: xj det [a_1 a_n] = det [a_1 a_j-1 c a_j+1 a_n] So: xj = det [a_1 a_j-1 c a_j+1 a_n] det [a_1 a_n] Want: b = [w x] such that [a b] [w x] = [1 0] [y z] [c d] [y z] [0 1]