Question

Show that each one of the following systems of linear equation is inconsistent:

x + y − 2z = 5

x − 2y + z = −2

−2x + y + z = 4

Answer 1

The above system can be written as:

`[(1, 1, -2),(1, -2, 1),(-2, 1, 1)][(x),(y),(z)] = [(5),(-2),(4)]`

or A X = B

`|A| = 1(-3) - 1(3) - 2 (-3) = -3 - 3 + 6 = 0`

So, A is singular. Now the system can be inconsistent, if

(adj A) × B ≠ 0

C₁₁ = −3   C₂₁ = −3   C₃₁ = −3

C₁₂ = −3   C₂₂ = −3   C₃₂ = −3

C₁₃ = −3   C₂₃ = −3   C₃₃ = −3

(adj A) = `[(-3, -3, -3),(-3, -3, -3),(-3, -3, -3)] = [(-3, -3, -3),(-3, -3, -3),(-3, -3, -3)]`

(adj A) × (B) = `[(-3, -3, -3),(-3, -3, -3),(-3, -3, -3)] [(5),(-2),(4)] = [(-15 + 6 - 12),(-15 + 6 -12),(-15 + 6 - 12)]`

= `[(-21),(-21),(-21)]`

≠ 0

Hence, the given system is inconsistent.