Question

The system of equations
$ x + 4y - 3z = 3 $
$ x - y + 7z= 11 $
$ 2x + 8y - 6z = 7 $
have

• A. unique solution
• B. infinitely many solutions
• C. no solutions
• D. only finite number of solutions

Answer 1

Answer: (C)

The given system of equations can be written in matrix form as,
$\begin{bmatrix}1&4&-3\\ 1&-1&7\\ 2&8&-6\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}3\\ 11\\ 7\end{bmatrix}$
i.e., $A X=B$
Now, $\left|A\right| = \begin{vmatrix}1&4&-3\\ 1&-1&7\\ 2&8&-6\end{vmatrix} = 0 $
$\therefore $ Solution is not unique.
Now, $adj\left(A\right) = \begin{bmatrix}-50&0&25\\ 20&0&-10\\ 10&0&-5\end{bmatrix}$
Now, $\left(adj A\right)B = \begin{bmatrix}-50&0&25\\ 20&0&-10\\ 10&0&-5\end{bmatrix}\begin{bmatrix}3\\ 11\\ 7\end{bmatrix}\ne O$
$\therefore $ The given system of equations have no solution.