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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
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Solution
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.
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