Question

Examine the consistency of the system of equations.
x+y+z=12x+3y+2z=2,
ax+ay+2az=4

Answer 1

The given system of equations is:
x + y + z = 1 2x + 3y + 2z = 2
ax + ay + 2az = 4
This system of equations can be written in the form 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=$\begin{bmatrix}1\\\2\\\4\end{bmatrix}$
Now,
IAI=1(6a-2a)-1(4a-2a)+1(2a-3a)
=4a-2a-a=4a-3a=a ≠0
∴ A is non-singular.
Therefore, A-1 exists.

Hence, the given system of equations is consistent.