Question

Check whether the following system of equations has a solution.
4x+2y+8z+6w = 3
2x+2y+2z+2w = 1
x+3z+2w = 3

Answer 1

Answer on Question #79914 - Math – Linear Algebra

Question

Check whether the following system of equations has a solution.

$$4x + 2y + 8z + 6w = 3$$ $$2x + 2y + 2z + 2w = 1$$ $$x + 3z + 2w = 3$$

Solution

Find a rank of the coefficient matrix A:

$$A = \begin{pmatrix} 4 & 2 & 8 & 6 \\ 2 & 2 & 2 & 2 \\ 1 & 0 & 3 & 2 \end{pmatrix} \sim \begin{pmatrix} 1 & 0.5 & 2 & 1.5 \\ 2 & 2 & 2 & 2 \\ 1 & 0 & 3 & 2 \end{pmatrix} \sim \begin{pmatrix} 1 & 0.5 & 2 & 1.5 \\ 0 & 1 & -2 & -1 \\ 0 & -0.5 & 1 & 0.5 \end{pmatrix} \sim \begin{pmatrix} 1 & 0.5 & 2 & 1.5 \\ 0 & 1 & -2 & -1 \\ 0 & 0 & 0 & 0 \end{pmatrix}.$$ $$\operatorname{Rank}(A) = 2.$$

Write an augmented matrix of the system of the equations and find her rank:

$$B = \begin{pmatrix} 4 & 2 & 8 & 6 & 3 \\ 2 & 2 & 2 & 2 & 1 \\ 1 & 0 & 3 & 2 & 3 \end{pmatrix} \sim \begin{pmatrix} 1 & 0.5 & 2 & 1.5 & 0.75 \\ 2 & 2 & 2 & 2 & 1 \\ 1 & 0 & 3 & 2 & 3 \end{pmatrix} \sim \begin{pmatrix} 1 & 0.5 & 2 & 1.5 & 0.75 \\ 0 & 1 & -2 & -1 & -0.5 \\ 0 & -0.5 & 1 & 0.5 & 2.25 \end{pmatrix} \sim$$ $$\sim \begin{pmatrix} 1 & 0.5 & 2 & 1.5 & 0.75 \\ 0 & 1 & -2 & -1 & -0.5 \\ 0 & 0 & 0 & 0 & 2 \end{pmatrix} \sim \begin{pmatrix} 1 & 0.5 & 2 & 1.5 & 0.75 \\ 0 & 1 & -2 & -1 & -0.5 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}.$$ $$\operatorname{Rank}(B) = 3.$$

$\operatorname{Rank}(A) \neq \operatorname{Rank}(B)$. By Kronecker-Capelli theorem, the given system of linear equations is incompatible, the system has no solution.

Answer: the system has no solution.

Answer provided by https://www.AssignmentExpert.com