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 on
A. $\mu $ only
B. $\lambda $ only
C. $\lambda $ and $\mu $ both
D. neither $\lambda $ nor $\mu $

Answer 1

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.