Find all values of a and b for which the system is inconsistent.

Question

Consider the system of equations with the following augmented matrix, in which a and b are real numbers.

(1 0 2 a^2 3| b)

(0 a b 0 1| a)

(0 0 0 a 0| b^2-1)

(a) Find all values of a and b for which the system is inconsistent. Explain your answer.

Answer 1

A system is inconsistent if there is no solution - i.e. if you get an equation 0=x where x is not 0. With that in mind:

The last row in the augmented matrix represents the equation: 0x_1+0x_2+0x_3+ax_4+0x_5=b^2-1. When will this be inconsistent? When a = 0 and b^2-1 is not 0 or solving for b: b cannot equal +/- 1.