Gaussian elimination

In Gaussian elimination, the objective is to transform a system of equations into a row-echelon (triangular) form so that it is much easier to solve. Observe the following system, which is already in row-echelon form, and find the value of the unknowns $x$, $y$ and $z$ using back-substitution (from bottom to top):

$\begin{cases} x + y + z = 9 \\ 2y - z = 5 \\ 3z = 9 \end{cases}$

At the end of Gaussian elimination, a student arrives at the following coefficient matrix (where the last column is the constant term):

$\left(\begin{array}{ccc|c} 1 & 2 & -1 & 4 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right)$

1. a) Write down the system of equations associated with this matrix.
2. b) Based on the last row, indicate whether the system is Consistent with a unique solution, Consistent with infinitely many solutions, or Inconsistent. Reason your answer.

Sometimes, to apply Gaussian elimination more efficiently, it is useful to swap the rows of the system. Given the following system:

$\begin{cases} 3x - 2y + z = 10 \\ x + y + z = 6 \\ 2y - 3z = -4 \end{cases}$

1. a) Explain why it would be convenient to swap the first equation ($E_1$) with the second ($E_2$) before starting operations.
2. b) Write down the resulting system after the swap and perform the first step of Gaussian elimination to eliminate $x$ from the new second equation.

Solve the following system of linear equations using Gaussian elimination. To make your work easier, notice that the variable $x$ has already been eliminated from the second equation:

$\begin{cases} x - y + z = 2 \\ y + 2z = 8 \\ 2x + y - z = 1 \end{cases}$

Solve the following system of equations step by step using Gaussian elimination:

$\begin{cases} x + y + z = 6 \\ x - y + z = 2 \\ x + y - z = 0 \end{cases}$

Hint: Subtract the first equation from the second and third equations to eliminate $x$ in a single step.

Solve the following system of linear equations using Gaussian elimination. Remember to organise the steps to obtain a triangular structure before calculating the values of the unknowns:

$$\begin{cases} x + 2y - z = 3 \\ 2x - y + z = 7 \\ x + y - 2z = -2 \end{cases}$$

Determine whether the following system has a solution or not, applying the Gauss method until you obtain the last row:

$\begin{cases} x + y + z = 3 \\ 2x + 2y + 2z = 10 \\ 3x - y + z = 2 \end{cases}$

Justify your answer by analyzing the relationship between the first and the second equation.

Analyse and solve the following system. If the system turns out to have infinitely many solutions, express the general solution using a parameter $\lambda$ (or $t$):

$$\begin{cases} x - 3y + 2z = 4 \\ 2x + y - 3z = 1 \\ 5x - 8y + 3z = 13 \end{cases}$$

Find the value of the parameter $k$ for which the following system of equations is inconsistent (that is, has no solution):

$$\begin{cases} x + y + z = 2 \\ 2x + 3y + z = 3 \\ 3x + 4y + kz = 10 \end{cases}$$

A small technology company is managing three artificial intelligence projects: Alpha, Beta, and Gamma. The total budget allocated for the three is $450.000$ €. It is known that the cost of project Gamma is equal to the sum of the cost of Alpha plus twice Beta. Furthermore, if we subtract the cost of Gamma from three times the cost of Alpha, we get exactly the cost of Beta plus $30.000$ €.

Set up a system of linear equations and use Gaussian elimination to determine the exact investment required for each project.

A young investor decides to diversify her savings of $12.000$ € into three different financial products: a savings account yielding $2\%$, an investment fund yielding $5\%$, and tech stocks with a return of $8\%$.

It is known that:

• The total annual interest earned is $600$ €.

• The amount invested in tech stocks is half the sum of the other two investments.

Set up a system of three equations with three unknowns and solve it using Gaussian elimination to find how much money she invested in each product.

An artisan workshop manufactures three types of lamps: model Alpha, model Beta and model Gamma.

• To manufacture an Alpha lamp, 20 minutes of assembly, 10 of painting and 10 of packaging are needed.

• For a Beta lamp, 10 minutes of assembly, 20 of painting and 10 of packaging.

• For a Gamma lamp, 10 minutes of assembly, 10 of painting and 20 of packaging.

If the workshop dedicated a total of $100$ hours today to assembly, $100$ hours to painting and $100$ hours to packaging, how many lamps of each type did they manufacture?

Note: Remember to convert all time units to the same measure (minutes or hours) before setting up the system.