# Q : 4         Examine the consistency of the system of equations.                 $\small x+y+z=1$                 $\small 2x+3y+2z=2$                 $\small ax+ay+2az=4$

Answers (1)

We have given the system of equations:

$\small x+y+z=1$

$\small 2x+3y+2z=2$

$\small ax+ay+2az=4$

The given system of equations can be written in the form of the matrix; $AX =B$

where $A = \begin{bmatrix} 1& 1&1 \\ 2& 3& 2\\ a& a &2a \end{bmatrix}$,  $X = \begin{bmatrix} x\\y \\ z \end{bmatrix}$  and $B = \begin{bmatrix} 1\\2 \\ 4 \end{bmatrix}$.

So, we want to check for the consistency of the equations;

$|A| = 1(6a-2a) -1(4a-2a)+1(2a-3a)$

$= 4a -2a-a = 4a -3a =a \neq 0$

[If zero then it won't satisfy the third equation]

Here A is non- singular matrix therefore there exist $A^{-1}$.

Hence, the given system of equations is consistent.

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