Discuss the solutions of the system of equations for all values of $\lambda$. $ x+y+z=2\;,2x+y-2z=2\;,\lambda\;x+y+4z=2$

Answer 1

Toolbox:

Step 1

The matrix equation is $ \begin{bmatrix} 1 & 1 & 1 \\ 2 & 1 & -2 \\ \lambda & 1 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix}$

$ AX = B$

The angmented matrix

$ [A,B] = \begin{bmatrix} 1 & 1 & 1 & 2 \\ 2 & 1 & -2 & 2 \\ \lambda & 1 & 4 & 2 \end{bmatrix} \sim \begin{bmatrix} 1 & 1 & 1 & 2 \\ 0 & -1 & -4 & -2 \\ 0 & 1-\lambda & 4-\lambda & 2\lambda \end{bmatrix} R_2 \rightarrow R_2-2R_1$

$ R_3 \rightarrow R_3- \lambda R_1$

$ \sim \begin{bmatrix} 1 & 1 & 1 & 2 \\ 0 & -1 & -4 & -2 \\ 0 & -\lambda & -\lambda & -2\lambda \end{bmatrix} R_3 \rightarrow R_3+R_2$

When $ \lambda = 0$ the last equivalent matrix becomes $ \begin{bmatrix} 1 & 1 & 1 & 2 \\ 0 & -1 & -4 & 2 \\ 0 & 0 & 0 & 0 \end{bmatrix}$ which is in echelon form.

There are two nonzero rows and $ \rho(A) = \rho(A,B)=2$

The system is consistent with infinitely many solutions.

When $ \lambda \neq 0, $ there are 3 nonzero rows and $ \rho(A)=\rho(A,B)=3$. The system has a unique solution.

Comments

In Step 1 (i) "angmented" to read as "augmented"
(ii) R3→R3−λR1 to be placed below R2→R2−2R1
(iii) R3→R3+R2 to be placed against row 3 of the matrix.
(iv) Step 2 begins just above "When λ=0..."
(v) Step # begins before "When λ≠0,..."