The set of all values of lambda for which the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The given system of equations can be rewritten as:$(1 - \lambda)x - 2y - 2z = 0$$x + (2 - \lambda)y + z = 0$$-x - y - \lambda z = 0$For the system

Vedclass · Accepted Answer

is a singleton

Vedclass · Answer

(A) The given system of equations can be rewritten as:$(1 - \lambda)x - 2y - 2z = 0$$x + (2 - \lambda)y + z = 0$$-x - y - \lambda z = 0$For the system to have non-zero solutions,the determinant of the coefficient matrix must be zero:$\begin{vmatrix} 1 - \lambda & -2 & -2 \ 1 & 2 - \lambda & 1 \ -1 & -1 & -\lambda \end{vmatrix} = 0$Expanding the determinant along the first row:$(1 - \lambda) [(2 - \lambda)(-\lambda) - (-1)(1)] - (-2) [1(-\lambda) - (-1)(1)] + (-2) [1(-1) - (-1)(2 - \lambda)] = 0$$(1 - \lambda) [-2\lambda + \lambda^2 + 1] + 2 [-\lambda + 1] - 2 [-1 + 2 - \lambda] = 0$$(1 - \lambda)(\lambda - 1)^2 + 2(1 - \lambda) - 2(1 - \lambda) = 0$$-(\lambda - 1)^3 = 0$$\lambda = 1$Since there is only one value for $\lambda$,the set of all values is a singleton set.

The set of all values of $\lambda$ for which the system of linear equations $x - 2y - 2z = \lambda x$,$x + 2y + z = \lambda y$,and $-x - y = \lambda z$ has non-zero solutions.

Similar Questions

$\left| {\begin{array}{ccc} 19 & 17 & 15 \\ 9 & 8 & 7 \\ 1 & 1 & 1 \end{array}} \right| = $

If the area of a triangle is $35$ sq. units with vertices $(2, -6)$,$(5, 4)$,and $(k, 4)$,then $k$ is . . . . . . .

If the system of equations $ax + y + z = 0$,$x + by + z = 0$ and $x + y + cz = 0$,where $a, b, c \neq 1$,has a non-trivial solution,then the value of $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c}$ is

If $A = \begin{bmatrix} \alpha^2 & 5 \\ 5 & -\alpha \end{bmatrix}$ and $\det(A^{10}) = 1024$,then $\alpha = $

The value of $\left| \begin{array}{ccc} a & a + b & a + 2b \\ a + 2b & a & a + b \\ a + b & a + 2b & a \end{array} \right|$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module