The system of linear equations lambda x + y + | vedclass

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(B) The augmented matrix $[A|B]$ is given by $\begin{bmatrix} \lambda & 1 & 1 & 3 \ 1 & -1 & -2 & 6 \ -1 & 1 & 1 & \mu \end{bmatrix}$.Performing row

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infinite number of solutions for $\lambda = -1$ and $\mu = 3$

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(B) The augmented matrix $[A|B]$ is given by $\begin{bmatrix} \lambda & 1 & 1 & 3 \ 1 & -1 & -2 & 6 \ -1 & 1 & 1 & \mu \end{bmatrix}$.Performing row operations:$R_1 \leftrightarrow R_2$ gives $\begin{bmatrix} 1 & -1 & -2 & 6 \ \lambda & 1 & 1 & 3 \ -1 & 1 & 1 & \mu \end{bmatrix}$.$R_3 ightarrow R_3 + R_1$ gives $\begin{bmatrix} 1 & -1 & -2 & 6 \ \lambda & 1 & 1 & 3 \ 0 & 0 & -1 & \mu + 6 \end{bmatrix}$.$R_2 ightarrow R_2 - \lambda R_1$ gives $\begin{bmatrix} 1 & -1 & -2 & 6 \ 0 & 1+\lambda & 1+2\lambda & 3-6\lambda \ 0 & 0 & -1 & \mu + 6 \end{bmatrix}$.For infinite solutions,the rank of the matrix must be less than the number of variables $(3)$.If $\lambda = -1$,the second row becomes $[0, 0, -1, 9]$.The system becomes consistent with infinite solutions if the equations are dependent,which occurs when $\mu = 3$.

The system of linear equations $\lambda x + y + z = 3$,$x - y - 2z = 6$,and $-x + y + z = \mu$ has:

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