Investigate for what values of $λ$ and $μ$ the system of equations $x+y+z=6$, $x+2y+3z=10$, $x+2y+λz=μ$ has a) No solutions b) Infinite number of solutions c) Unique solution
How to do this question with Augmented Matrix method can somebody please explain fast I have a test in 4 hours :(
My row echelon form is coming $\left[ \begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 1 & 2 & 3 & 10 \\ 1 & 2 & \lambda & \mu \end{array} \right]$
$\left[ \begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 1 & 2 & 3 & 10 \\ 1 & 2 & \lambda & \mu \end{array} \right]\sim\left[ \begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 1 & 2 & 3 & 10 \\ 0 & 0 & \lambda-3 & \mu-10 \end{array} \right]\sim\left[ \begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 0 & \lambda-3 & \mu-10 \end{array} \right]$
For $\lambda\neq 3$ this will have a unique solution for all $\mu$, as we can reduce the matrix into $I$ and thus find the solution on the right side of the augmented matrix.
For $\lambda=3$ and $\mu=10$, then there are infinitely many (the last row is completely $0$, just look what happens in the system of equations). If $\mu\neq 10$, there are no solutions, again take a look at the system of equations.