Question

3. (5 points) Is the system -21 +2r2 +6c3 31 -22-83 34-557 consistent? Which variables are free? Which variables are dependent? Solve the system. 4. (5 points) Find a value of k which will make the system x1 + x2- x3 = 1 21 2 + 23 3 inconsistent. Hint: Perform row reduction on the augmented matriz. What condition on k will force the last column to be a pivot column?

Answer 1

Augmented matrix is [1 - 2 3 - 1 2 - 2 - 3 6 - 8 1 0 3 - 1 - 6 - 5 - 2 - 6 - 7] Applying elementary row operation we get, [1 - 2 3 - 1 2 - 2 - 3 6 - 8 1 0 3 - 1 - 6 - 5 - 2 - 6 - 7] R_2 = R_2 + 2R_1 R_3 = R_3 - 3R_1 [1 0 0 - 1 0 1 - 3 0 1 1 2 0 - 1 - 8 - 2 - 2 - 10 - 1] R_2 left- rightarrow R_3 [1 0 0 - 1 1 0 - 3 1 0 1 0 2 - 1 - 2 - 8 - 2 - 1 - 10] x_1 x_2 x_3 x_4 x_5 There is no pivot in the last column Therefore the given system is consistent. We see that x_3 and x_5 are free and x_1, x_2, x_4 are dependent. Let, x_3 = s, x_5 = t Therefore x_1 - x_2 - 3x_3 + x_4 - x_5 =