Linear systems and rank

Solve with Gaussian elimination the system

$$\{\begin{array}{l}x+y+z=3\\ x+2y-z=0\\ 2x-y+3z=10\end{array}$$

Solution available to those who have unlocked the subject.

Solve with Gaussian elimination the system

$$\{\begin{array}{l}x-y+z=2\\ 2x+y-z=1\\ x+2y-2z=-1\end{array}$$

Solution available to those who have unlocked the subject.

Solve with Gaussian elimination the system in the unknowns $x_1,x_2,x_3,x_4$

$$\{\begin{array}{l}{x}_1-x_2+2x_3+x_4=1\\ 2x_1-2x_2+5x_3+3x_4=4\end{array}$$

Solution available to those who have unlocked the subject.

Determine whether the system

$$\{\begin{array}{l}x+2y-z=1\\ 2x+3y+z=4\\ 3x+5y=6\end{array}$$

has solutions.

Solution available to those who have unlocked the subject.

Compute the rank of the matrix

$$A=\left(\begin{array}{cccc}1& -1& 2& 1\\ 2& 1& 1& 0\\ 1& 2& -1& -1\end{array}\right).$$

Solution available to those who have unlocked the subject.

Compute the rank of the matrix

$$A=\left(\begin{array}{cccc}1& 1& 2& k\\ 1& -1& 0& 1\\ 2& 0& 2& 3\end{array}\right)$$

as $k\in \mathbb{R}$ varies.

Solution available to those who have unlocked the subject.

Determine with the Rouché-Capelli theorem whether the system

$$\{\begin{array}{l}{x}_1+x_2+x_3+x_4=2\\ x_1-x_2+x_3-x_4=0\\ 2x_1+2x_3=2\end{array}$$

is consistent, how many solutions it has and what they are.

Solution available to those who have unlocked the subject.

Solve with Cramer’s rule the system

$$\{\begin{array}{l}x+y+z=2\\ 2x-y+z=5\\ x+2y-z=-3\end{array}$$

Solution available to those who have unlocked the subject.

Find the value of the unknown $x$ in the solution of the system

$$\{\begin{array}{l}2x+y+z=1\\ x+3y-z=2\\ x-y+2z=0\end{array}$$

Solution available to those who have unlocked the subject.

Find a basis and the dimension of the solution space of the homogeneous system

$$\{\begin{array}{l}{x}_1+x_2+x_3+x_4=0\\ x_1-x_2+2x_3=0\end{array}$$

Solution available to those who have unlocked the subject.

Find the solutions of the system

$$\{\begin{array}{l}x+2y-z=3\\ 2x+4y+z=0\end{array}$$

and write them as a particular solution plus solutions of the associated homogeneous system.

Solution available to those who have unlocked the subject.

Discuss, as $k\in \mathbb{R}$ varies, the system

$$\{\begin{array}{l}x+y+z=1\\ x+2y+kz=2\\ x+ky+2z=k\end{array}$$

Solution available to those who have unlocked the subject.

Find the values of $k\in \mathbb{R}$ for which the system

$$\{\begin{array}{l}x+2y=1\\ x-y=4\\ 2x+y=k\end{array}$$

is consistent and, for those values, solve it.

Solution available to those who have unlocked the subject.