Gauss's method
You already know Gauss's method from 1st Bachillerato: triangularising a system by combining rows until it is in echelon form. Now that you have the language of matrices and rank, Gauss's method becomes something more powerful: it is, in fact, the procedure you use to calculate the rank of any matrix, and therefore the one that lets you apply the Rouché-Frobenius theorem in practice, even with more equations than unknowns.
Triangularising is calculating the rank
Classify the following system according to its number of solutions after applying Gaussian elimination. State whether it is consistent with a unique solution, consistent…
Classify the following system according to its number of solutions after applying Gaussian elimination. State whether it is consistent with a unique solution, consistent with infinitely many solutions, or inconsistent, based on the last row obtained:
$$\begin{cases} x + 2y - z = 3 \\ 2x + 4y - 2z = 1 \\ x + y + z = 2 \end{cases}$$Systems with a parameter: triangularising with a letter
Consider the following system of linear equations depending on the real parameter $k$: $\begin{cases} kx + y + z = k^2 \\ x + ky + z = k \\ x + y + kz = 1 \end{cases}$…
Consider the following system of linear equations depending on the real parameter $k$:
$\begin{cases} kx + y + z = k^2 \\ x + ky + z = k \\ x + y + kz = 1 \end{cases}$
- a) Discuss the system using the Gaussian elimination method according to the different values of $k$.
- b) In the case where the system is consistent dependent (underdetermined), determine the general solution and find a particular solution such that the sum of its components is equal to 1.
- c) Is there any value of $k$ for which the system represents three planes intersecting in a line? Justify your answer.
Consider the system of equations: $\begin{cases} x + ay + z = a \\ ax + y + z = 1 \\ x + y + az = a^2 \end{cases}$ a) Discuss the system for the different real values of…
Consider the system of equations:
$\begin{cases} x + ay + z = a \\ ax + y + z = 1 \\ x + y + az = a^2 \end{cases}$
- a) Discuss the system for the different real values of $a$.
- b) If $a = 1$, describe the solution geometrically.
- c) If $a = -2$, show using Gaussian elimination that the system is inconsistent. What geometric configuration do the three planes form in this case? Justify your answer with the relative positions of the planes (for example, by studying their normal vectors) and use a sketch drawn by yourself to help.
More equations than unknowns
Study the following system of 4 equations in 3 unknowns using Gaussian elimination:…
Study the following system of 4 equations in 3 unknowns using Gaussian elimination:
$\begin{cases} x + y + z = 2 \\ 2x + 3y + z = 3 \\ ax + 10y + 4z = 11 \\ 3x + 4y + 2z = a \end{cases}$
- a) Determine for which value of $a$ the system is consistent.
- b) For this value of $a$, solve the system and indicate whether the solution is unique.
- c) What relative position do the four planes defined by the equations occupy when the system is consistent?
Gauss-Jordan: zeros on both sides
In a smart power grid, the current intensities $I_1, I_2, I_3$ and $I_4$ at four nodes satisfy the following Kirchhoff's laws, scaled by a resistance factor $R$:…
In a smart power grid, the current intensities $I_1, I_2, I_3$ and $I_4$ at four nodes satisfy the following Kirchhoff's laws, scaled by a resistance factor $R$:
$\begin{cases} I_1 + I_2 - I_3 + I_4 = 0 \\ 2I_1 - I_2 + I_3 = 5 \\ I_1 + I_3 - 2I_4 = 1 \\ 3I_1 + I_4 = 6 \end{cases}$
- a) Solve the system using the Gauss–Jordan elimination method (achieving the maximum possible number of zeros in the matrix).
- b) Is the system consistent with a unique solution? Justify whether it is possible to remove one of the measurements without losing information about the currents.
Designing and checking an elimination algorithm
The algorithm must also work if the first candidate for a pivot is zero. Before dividing, look for a lower row with a non-zero entry in that column and swap it in. If the whole remaining column is zero, move on to the next column without increasing the number of pivots.
fila_pivote ← 0
para cada columna de coeficientes:
buscar desde fila_pivote una fila con entrada no nula
si no existe: continuar con la siguiente columna
intercambiar esa fila con fila_pivote
para cada fila inferior:
factor ← entrada / pivote
restar factor * fila_pivote a toda la fila (incluido B)
fila_pivote ← fila_pivote + 1
examinar filas nulas y términos independientes
Choose between these outcomes after eliminating:
*Unique solution if there are n pivots in the coefficients and no inconsistent row. With fewer than n pivots and no inconsistency there are free variables.
Modify the pseudocode so that each pivot is turned into 1 and the entries above it are also eliminated: you obtain Gauss-Jordan. Run it by hand or program it, compare the number of operations and check the three tests with a matrix calculation tool. Hand in the intermediate matrices and check the solutions in the original equations. With decimal data, preferably select the available pivot with the largest absolute value and state a tolerance for deciding what counts as approximately zero; exact ranks with parameters require separate algebraic cases.
Reviewing the process as a team
Rotate the tasks of designing, carrying out, checking and explaining. Each member predicts the type of solution before calculating. If a discrepancy appears, find the first row that differs and reconstruct the operation without blaming anyone's ability. If you get stuck, write down what you know, set aside a simple case and ask for a specific hint. Hand in one corrected error, the strategy that helped you overcome it and an alternative proposed by another member. Compare methods with arguments, not by who finishes first.