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Question
Examine the consistency of the system of equations.
x + 3y = 5
2x + 6y = 8
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Solution
Let, A = `[(1,3),(2,6)]`, X = `[(x),(y)]`, B = `[(5),(8)]`
Then the given system of equations can be written as,
`[(1,3),(2,6)][(x),(y)] = [(5),(8)]`
Now, |A|
= `|(1,3),(2,6)|`
= 1 × 6 − 2 × 3
= 6 − 6
= 0
Cofactors of the elements of |A| are, respectively,
A11 = 6, A12 = −2, A21 = −3, A22 = 1
∴ adj A = `[(6,-2),(-3,1)] = [(6,-3),(-2,1)]`
⇒ (adj A)B = `[(6,-3),(-2,1)] [(5),(8)]`
= `[(30 - 24),(-10 + 8)]`
= `[(6),(-2)] ≠ 0`
|A| = 0 and (adj A)B ≠ 0
Therefore, the given system of equations is inconsistent.
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