Question

# The existence of the unique solution of the system for equations:$x + y + z = \lambda \\ 5x - y + \mu z = 10 \\ 2x + 3y - z = 6 \\$depends onA. $\mu$ onlyB. $\lambda$ onlyC. $\lambda$ and $\mu$ bothD. neither $\lambda$ nor $\mu$

Hint: Express the given system of equations in matrix form and find the determinant of the coefficients of x,y and z.

We will write given equations in the matrix form as $A.X = B$
Where $A = \left( {\begin{array}{*{20}{c}} 1&1&1 \\ 5&{ - 1}&\mu \\ 2&3&{ - 1} \end{array}} \right)$ , $X = \left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right)$ and $B = \left( {\begin{array}{*{20}{c}} \lambda \\ {10} \\ 6 \end{array}} \right)$
Now, we will find determinant of A i.e. $\left| A \right|$
$\left| A \right| = \left( {\begin{array}{*{20}{c}} 1&1&1 \\ 5&{ - 1}&\mu \\ 2&3&{ - 1} \end{array}} \right) \\ \left| A \right| = 1\left( {1 - 3\mu } \right) - 1\left( { - 5 - 2\mu } \right) + 1\left( {15 + 2} \right) \\ \left| A \right| = 1 - 3\mu + 5 + 2\mu + 17 \\ \left| A \right| = 23 - \mu \\$
From the above equation, we can see that the uniqueness of the system depends only on $\mu$.
$\therefore$Correct option is A.

Note: In a practical case, a system of linear equations will have a unique solution if the lines
representing the equations intersect each other at only one unique point i.e. the lines are
neither parallel nor coincident.