JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Matrices

For a system of equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero, and the determinant of the augmented matrix must also be zero.

First, set the coefficient determinant to zero:


Expanding: $2(25-2\lambda )-(-10-\lambda )-(4+5)=0$

Simplifying: $50-4\lambda +10+\lambda -9=0\Rightarrow 51=3\lambda \Rightarrow \lambda =17$

Next, set the augmented determinant to zero:


Expanding: $2(-35-2\mu )-(14-\mu )+5(4+5)=0$

Simplifying: $-70-4\mu -14+\mu +45=0\Rightarrow -3\mu =39\Rightarrow \mu =-13$

Finally: $(\lambda +\mu {)}^2+(\lambda -\mu {)}^2=2({\lambda }^2+{\mu }^2)=2(17^2+13^2)=2(289+169)=916$