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  • I have a doubt, kindly clarify. - The system of linear equations - Matrices and Determinants - JEE Main

The system of linear equations 

\\ x + y +z =2 \\ 2x + 3y + 2z = 5 \\ 2x+ 3y + (a^2 -1)z = a +1

  • Option 1)

     

    is inconsistent when a = 4

  • Option 2)

    has a unique solution for |a| = \sqrt3

  • Option 3)

     

    has infinitely many solutions for a = 4

  • Option 4)

     

    is inconsistent when |a| = \sqrt3

Answers (1)

best_answer

 

Inconsistent system of linear equation -

If the system of equations has no solutions

-

 

from the concept of cramer's rule 

D=\begin{vmatrix} 1 &1 &1 \\ 2&3 &2 \\ 2&3 & a^{2}-1 \end{vmatrix}=a^{2}-3

D_{1}=\begin{vmatrix} 2 &1 &1 \\ 5&3 &2 \\ a+1&3 & a^{2}-1 \end{vmatrix}=a^{2}-a+1

D_{2}=\begin{vmatrix} 1 &2 &1 \\ 2&5 &2 \\ 2&a+1 & a^{2}-1 \end{vmatrix}=a^{2}-3

D_{3}=\begin{vmatrix} 1 &1 &2 \\ 2&3 &5 \\ 2&3 & a+1 \end{vmatrix}=a-4

Now, D=0\: \: \: \: \: @\: \: \: \: \: \left | a \right |=\sqrt{3}

but   D_{3}=\pm \sqrt{3}-4\neq 0

So the system is inconsistant for \left | a \right |=\sqrt{3}.


Option 1)

 

is inconsistent when a = 4

Option 2)

has a unique solution for |a| = \sqrt3

Option 3)

 

has infinitely many solutions for a = 4

Option 4)

 

is inconsistent when |a| = \sqrt3
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