Examine the consistency of the system of equations

$x + y + z = 1, 2 x + 3 y + 2 z = 2, a x + a y + 2 a z = 4$

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Solution

The given system can be written as AX=B, where
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 2 & 3 & 2 a & a & 2 a \end{matrix} ⎤ ⎥ ⎦ . X = ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ B = ⎡ ⎢ ⎣ \begin{matrix} 124 \end{matrix} ⎤ ⎥ ⎦$
Here, $| A | = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 2 & 3 & 2 a & a & 2 a \end{matrix} ∣ ∣ ∣ ∣ = 1 (6 a - 2 a) - 1 (4 a - 2 a) + 1 (2 a - 3 a)$
$= 4 a - 2 a - a = 4 a - 3 a = a \neq 0$
$∴$ A is non-singular. Therefore, $A^{- 1}$ exists.
Hence, the given system of equations is consistent.

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