Matrices, determinants and linear systems

Given the matrices

$$A=\left(\begin{array}{ccc}1& -2& 0\\ 3& 1& 2\end{array}\right),B=\left(\begin{array}{cc}2& 1\\ 0& -1\\ 1& 3\end{array}\right),$$

compute the products $AB$ and $BA$.

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Two customers buy three goods. The first buys $2$ units of the first good, $1$ of the second and $3$ of the third; the second buys $0$, $4$ and $1$ units. The unit prices are $5$ euros, $3$ euros and $2$ euros. Write the matrix $Q$ of the quantities (one row per customer) and the column vector $p$ of the prices, and compute with the product $Qp$ the expenditure of each customer.

Solution available to those who have unlocked the subject.

Compute the determinant

$$\left|\begin{array}{ccc}2& -1& 3\\ 1& 0& 2\\ 4& 1& -1\end{array}\right|.$$

Solution available to those who have unlocked the subject.

Compute the determinant

$$\left|\begin{array}{cccc}2& 1& 0& 3\\ 0& 1& 0& 0\\ 1& 3& 2& 1\\ 0& 2& 0& 1\end{array}\right|.$$

Solution available to those who have unlocked the subject.

Compute the inverse of the matrix

$$A=\left(\begin{array}{cc}4& 3\\ 3& 2\end{array}\right).$$

Solution available to those who have unlocked the subject.

Compute the inverse of the matrix

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

Solution available to those who have unlocked the subject.

Find for which values of the real parameter $k$ the matrix

$$A=\left(\begin{array}{ccc}k& 2& 1\\ 2& k& 1\\ 1& 1& 1\end{array}\right)$$

is invertible.

Solution available to those who have unlocked the subject.

Find the rank of the matrix

$$A=\left(\begin{array}{ccc}1& 1& 2\\ 2& k& 4\\ 3& 3& k+4\end{array}\right)$$

as the real parameter $k$ varies.

Solution available to those who have unlocked the subject.

Solve with Cramer’s rule the system

$$\{\begin{array}{l}2x+y-z=3\\ x-y+2z=1\\ 3x+2y+z=8\end{array}$$

Solution available to those who have unlocked the subject.

Discuss with the Rouché-Capelli theorem and solve with Gauss’s method the system

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

Solution available to those who have unlocked the subject.

Discuss as the real parameter $k$ varies the system

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

and solve it when it is consistent.

Solution available to those who have unlocked the subject.

The demand and supply functions of two goods are

$$q_{d1}=24-3p_1+p_2,q_{s1}=2p_1-2,q_{d2}=10+p_1-2p_2,q_{s2}=3p_2-4.$$

Find the equilibrium prices and quantities.

Solution available to those who have unlocked the subject.

An economy with two sectors has technology matrix and final demand

$$A=\left(\begin{array}{cc}0.3& 0.2\\ 0.1& 0.4\end{array}\right),d=\left(\begin{array}{c}20\\ 40\end{array}\right).$$

Find the production levels of the two sectors that meet the final demand.

Solution available to those who have unlocked the subject.

A firm produces three items. One unit of the first requires $2$ hours in department A, $1$ in department B and $1$ in department C; one unit of the second requires $1$, $3$ and $1$ hours; one unit of the third requires $1$, $2$ and $3$ hours. Departments A, B and C have $80$, $130$ and $80$ hours available. Find the quantities of the three items that use up all the available hours.

Solution available to those who have unlocked the subject.