If the system of linear equations: x + y + z | vedclass

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

Question

(C) For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero, and the augmented matr

Vedclass · Accepted Answer

$22$

Vedclass · Answer

(C) For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero, and the augmented matrix must be consistent.First, calculate the determinant $\Delta$ of the coefficient matrix:$\Delta = \begin{vmatrix} 1 & 1 & 1 \ 1 & 2 & 5 \ 2 & 3 & \lambda \end{vmatrix} = 1(2\lambda - 15) - 1(\lambda - 10) + 1(3 - 4) = 0$$2\lambda - 15 - \lambda + 10 - 1 = 0$$\lambda - 6 = 0 \Rightarrow \lambda = 6$.Now, consider the augmented matrix $[A|B] = \begin{pmatrix} 1 & 1 & 1 & | & 6 \ 1 & 2 & 5 & | & 10 \ 2 & 3 & 6 & | & \mu \end{pmatrix}$.Perform row operations: $R_2 	o R_2 - R_1$ and $R_3 	o R_3 - 2R_1$:$\begin{pmatrix} 1 & 1 & 1 & | & 6 \ 0 & 1 & 4 & | & 4 \ 0 & 1 & 4 & | & \mu - 12 \end{pmatrix}$.For infinitely many solutions, the last row must be a zero row, so $\mu - 12 = 4 \Rightarrow \mu = 16$.Thus, $\lambda = 6$ and $\mu = 16$.The value of $\lambda + \mu = 6 + 16 = 22$.

If the system of linear equations: $x + y + z = 6, x + 2y + 5z = 10, 2x + 3y + \lambda z = \mu$ has infinitely many solutions, then the value of $\lambda + \mu$ equals:

Similar Questions

The existence of the unique solution of the system $x + y + z = \lambda$,$5x - y + \mu z = 10$,and $2x + 3y - z = 6$ depends on

If $x=\alpha, y=\beta, z=\gamma$ is the unique solution of the system of linear equations $2x-3y+5z=12$, $5x+2y+3z=11$, and $x+2y-3z=-3$, then $2\alpha+5\beta+3\gamma=$

If $3X + 2Y = I$ and $2X - Y = O$,where $I$ and $O$ are unit and null matrices of order $3$ respectively,then

Consider the simultaneous linear equations $AX=B$ and $AY=Q$. If $A$ is an invertible matrix and $B$ is the unique solution of $AY=Q$, then the solution of $AX=B$ is

The set of values of $k$ for which the system of simultaneous equations $x+y+kz=1$,$2x+2y=3$,and $x+2y+2kz=k$ has no real solution is

Vedclass Test Series

Exam Paper Generator

Online Exam Module