Exercises: Gaussian elimination

A small video game shop tracks its sales of two titles ($V_1$ and $V_2$) during the first two days of the week. The data are organised in the following matrix $A$:

$A = \begin{pmatrix} 12 & 5 \\ 8 & 4 \end{pmatrix}$

Calculate the determinant of the matrix $A$ to check whether the sales vectors are linearly independent.

Calculate the value of the following determinant of order 3 using Sarrus' rule:

$\Delta = \begin{vmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 1 & 4 & 2 \end{vmatrix}$

Without carrying out any complex calculations, explain why the determinant of the matrix $B$ is equal to zero by applying the properties of determinants:

$B = \begin{pmatrix} 2 & -3 & 5 \\ 4 & -6 & 10 \\ 1 & 1 & 1 \end{pmatrix}$

Given the following matrix $M$, perform the elementary row operation $F_2 = F_2 - 2F_1$ to obtain a zero in element $a_{21}$ and write down the new resulting matrix. Does the value of the determinant change when performing this operation?

$M = \begin{pmatrix} 1 & 3 \\ 2 & 10 \end{pmatrix}$

Gaussian elimination allows us to calculate determinants by transforming the matrix into an "upper triangular" one. Calculate the determinant of the following matrix $T$, knowing that it is already in echelon form (it is upper triangular):

$T = \begin{pmatrix} 5 & -2 & 4 \\ 0 & 3 & 1 \\ 0 & 0 & -2 \end{pmatrix}$

The figure highlights the main diagonal and the zeros below it.

Use Gaussian elimination to calculate the determinant of the matrix $A$. To do this, make zeros in the first column below the first entry:

$|A| = \begin{vmatrix} 1 & 2 & 1 \\ 3 & 4 & 0 \\ 0 & 1 & 2 \end{vmatrix}$

Determine for which value of the parameter $k$ the determinant of the following matrix is zero, using Gaussian elimination to simplify the matrix:

$M = \begin{pmatrix} 1 & 5 \\ 2 & k \end{pmatrix}$

Apply Gaussian elimination to transform the following determinant into that of a triangular matrix and find its final value:

$|D| = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 5 \end{vmatrix}$

Marcos has to calculate the determinant of the matrix $A = \begin{pmatrix} 2 & 5 \\ 3 & 1 \end{pmatrix}$ applying the Gaussian elimination method. In his exam, he carries out the following working:

"To avoid working with fractions in the pivot of the first column, I apply the elementary row operation $F_2 \to 2F_2 - 3F_1$. I thus obtain the matrix $\begin{pmatrix} 2 & 5 \\ 0 & -13 \end{pmatrix}$, which is already upper triangular. Therefore, the determinant of the original matrix is $|A| = 2 \cdot (-13) = -26$".

His teacher tells him that the procedure contains a serious conceptual error.

1. a) Identify and explain with reasoning the error made by Marcos regarding how the elementary operation performed affects the value of the determinant.
2. b) Correct his reasoning to obtain the true value of $|A|$, keeping Marcos's idea of not introducing fractions during the triangularisation.

Consider the coefficient matrix $M = \begin{pmatrix} 1 & 2 & 1 \\ 3 & 10 & 5 \\ 2 & 6 & 11 \end{pmatrix}$, associated with a transport cost model:

1. a) Before calculating the determinant, notice that the elements of the main diagonal are strictly positive and dominant in magnitude over those in their row, and that there are no proportional rows. Based on these observations, make a justified estimate as to whether the determinant $|M|$ will turn out to be zero, strictly positive, or strictly negative.
2. b) Apply the Gaussian elimination method using exclusively transformations of the type $F_i \to F_i + kF_j$ (which do not change the determinant) to transform $M$ into an upper triangular matrix and calculate the exact value of $|M|$.
3. c) Check whether the sign obtained agrees with your estimate. If a classmate claimed that the determinant can be estimated simply by multiplying the original diagonal ($1 \cdot 10 \cdot 11 = 110$), explain in light of the performed Gaussian operations why that estimate overestimates the actual result.

Calculate the value of the following determinant of order 4 by first transforming it into an upper triangular matrix using Gaussian elimination (making zeros below the main diagonal):

A logistics company organises its routes using a cost matrix $M$. For the system to be efficient, the determinant of this matrix must not be zero. Apply row operations (Gaussian elimination) to determine whether the following matrix is non-singular (its determinant is non-zero):

What linear dependence relationship do you observe between its rows after carrying out the first step of Gaussian elimination?

Given the matrix $A$, use the properties of determinants and Gaussian elimination to obtain an equivalent determinant that has a row with two zeros, and calculate its final value:

In the study of a financial market, a matrix depending on a parameter $k$ has been obtained. Using Gauss's method to simplify the determinant, find the values of $k$ for which the rank of the matrix is less than 3:

Given that the determinant of a matrix $3 \times 3$ is $|B| = 12$, use row operations and the properties of determinants to calculate, with justification, the value of $|B'|$, where $B'$ is the matrix resulting from adding 5 times the first row to the third row of $B$.

Calculate the rank of the following matrix using Gaussian elimination (row echelon form). Explain how many linearly independent rows remain at the end of the process:

A graphic designer uses matrices to transform shapes. The "distortion" of the shape depends on the determinant of the transformation matrix. Apply Gaussian elimination to calculate the determinant of the following matrix of order 3 and check whether the shape maintains its area (this occurs if the determinant is 1 or -1):

Solve the following equation, using Gaussian elimination to simplify the determinant of the left-hand side before expanding:

To analyse the equilibrium of four interconnected economic sectors, calculating the determinant of the following matrix is proposed:

A work team proposes three different approaches:

• Strategy 1: Apply Sarrus' rule directly to the $4 \times 4$ matrix.

• Strategy 2: Expand along the elements of the first column using Laplace's formula (cofactors).

• Strategy 3: Triangularise the matrix using Gaussian elimination by combining elementary row operations.

1. a) Discard, in a justified manner, the strategy that is mathematically incorrect or not applicable to this matrix.
2. b) Compare the computational workload (number of minor determinants to calculate versus the number of row combinations required) between the two valid strategies and justify which one is more efficient for this case.
3. c) Apply the most efficient strategy you have chosen to calculate $|B|$, explaining in detail the effect on the sign of the determinant of each elementary operation you use.

A student started triangularising a matrix $A$ of order 3 to calculate its determinant using Gaussian elimination, but part of their notes has been erased. In their notebook, only the following can be clearly read:

1st) Incomplete original matrix: $A = \begin{pmatrix} 1 & 2 & -1 \\ 2 & x & 1 \\ -1 & 1 & y \end{pmatrix}$, where $x$ and $y$ are real numbers.

2nd) Successive elementary operations performed: first $F_2 \to F_2 - 2F_1$ and $F_3 \to F_3 + F_1$; then, on the resulting matrix, the operation $F_3 \to F_3 - F_2$.

3rd) Final upper triangular matrix obtained: $T = \begin{pmatrix} 1 & 2 & -1 \\ 0 & 3 & 3 \\ 0 & 0 & 2 \end{pmatrix}$.

1. a) Reconstruct, with reasoning, the values of the unknowns $x$ and $y$ of the original matrix from the described elementary operations.
2. b) Justify, stating the properties of the determinant associated with the Gaussian operations used, what the exact value of $|A|$ is.
3. c) If in the process the student had replaced the second row using the operation $F_2 \to 3F_2 - 2F_1$ instead of $F_2 \to F_2 - 2F_1$, what would the determinant of the resulting final triangular matrix have been? Justify it without needing to redo all the calculations.

Consider the matrix of order 4 depending on a real parameter $k$:

$M = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1+k & 1 & 1 \\ 1 & 1 & 1+k^2 & 1 \\ 1 & 1 & 1 & 1+k^3 \end{pmatrix}$

Use Gaussian elimination (triangulation) to transform the matrix and calculate its determinant in terms of $k$. Subsequently, discuss for which values of $k$ the matrix does not have an inverse.

Given the square matrices $A$ and $B$ of order 3 such that $|A| = -2$ and $|B| = 5$, solve the following equation to determine the value of the determinant of the unknown matrix $X$:

$| (2 \cdot A^t \cdot X)^{-1} \cdot B | = \frac{1}{10}$

Note: You must apply the properties of determinants (determinant of the inverse, of the transpose, of scalar multiplication and of matrix multiplication) to isolate $|X|$.

Find the rank of the following matrix $A$ depending on the values of the real parameters $a$ and $b$, using Gaussian elimination:

$A = \begin{pmatrix} 1 & a & 1 & 2 \\ 2 & b & 2 & 4 \\ 1 & a & a & 2 \end{pmatrix}$

Describe in detail all possible situations for the rank of the matrix depending on the relationships between $a$ and $b$.

A logistics company organizes the distribution of goods between four main nodes. If the connectivity matrix of the system is $A = \begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{pmatrix}$, prove, using Gaussian elimination, that its determinant is zero, and interpret what this result means regarding the linear dependence of the transport routes (rows of the matrix).

Find the inverse matrix of $A$ using the Gauss-Jordan algorithm (transforming $(A|I)$ into $(I|A^{-1})$):

$A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 2 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 4 & 1 \end{pmatrix}$

Hint: First perform the product of the two triangular matrices and then apply Gauss-Jordan to the resulting matrix.

In a sociological model of class mobility, the matrix $P$ represents the transition probability between three social strata in one generation. It is known that the system reaches an equilibrium when the matrix equation $(P^t - I) \cdot X = 0$ is satisfied, where $I$ is the identity matrix.

Given $P = \begin{pmatrix} 0,8 & 0,1 & 0,1 \\ 0,2 & 0,7 & 0,1 \\ 0,2 & 0,2 & 0,6 \end{pmatrix}$:

1. a) Calculate the determinant of the matrix $M = P^t - I$ using Gaussian elimination.
2. b) From the previous result, justify whether the rows of $M$ are linearly independent and what this implies for the rank of the matrix.

Find the value of the following determinant of order 4 by applying the properties of rows/columns and reducing it to a triangular determinant using Gaussian elimination:

$D = \begin{vmatrix} a & b & c & d \\ a & a & b & c \\ a & a & a & b \\ a & a & a & a \end{vmatrix}$

Express the final result in factorised form in terms of $a$, $b$, $c$ and $d$.

Determine for which values of the real parameter $m$ the rank of the following matrix is exactly 2, using Gaussian elimination:

$B = \begin{pmatrix} 1 & m & m^2 \\ 1 & 1 & 1 \\ 3 & m+2 & m^2+2 \end{pmatrix}$

For the values found, state which row is a linear combination of the other two.

The National Statistics Institute (INE) models the distribution of tourist expenditure (in hundreds of thousands of euros) across three Spanish autonomous communities—Andalucía ($C_1$), Cataluña ($C_2$), and Islas Baleares ($C_3$)—according to three booking channels. The coefficient matrix of the model depends on a seasonal index $a \in \mathbb{R}$:

To ensure that the allocation model has a unique and independent solution, the matrix must be non-singular ($|M(a)| \ne 0$).

1. a) Apply Gaussian elimination using transformations of the type $F_i \to F_i + kF_j$ to triangularise the matrix $M(a)$ and obtain the factorised algebraic expression of its determinant in terms of $a$.
2. b) Determine for which values of the seasonal index $a$ the model loses regularity (the determinant becomes zero).
3. c) For the specific case $a = 0$, interpret what happens among the three autonomous communities: prove the exact linear dependence relationship that exists between the rows of $M(0)$ using only the coefficients of the Gaussian operations you applied in part a).

When studying the invertibility of a matrix of order 4 depending on a real parameter $k$, Clara sets out to calculate its determinant using Gaussian elimination:

To do so, she applies the operations $F_2 \to F_2 - F_1$, $F_3 \to F_3 - 2F_1$, and $F_4 \to F_4 - F_1$, obtaining:

Next, Clara writes in her exam:

"To finish Gauss and get ones on the pivots, I need to divide $F_2$ by $(k-1)$ and $F_3$ by $k$. Therefore, if $k = 1$ or $k = 0$, Gaussian elimination cannot be applied and the determinant cannot be calculated in this way. For $k \notin \{0, 1\}$, the determinant is the product of the diagonal: $|A(k)| = k(k-1)^2$, so the matrix is invertible only if $k \ne 0$ and $k \ne 1$".

1. a) Identify and explain with reasoning the two conceptual errors made by Clara: one regarding the need to obtain ones on the diagonal to calculate determinants and another regarding the validity of Gaussian elimination in the cases $k = 0$ and $k = 1$.
2. b) Prove formally that $T(k)$ is an upper triangular matrix for any real value of $k$ and justify why the formula $|A(k)| = k(k-1)^2$ is valid for all $k \in \mathbb{R}$, including $k = 0$ and $k = 1$.
3. c) If before triangularising, columns 2 and 3 had been swapped ($C_2 \leftrightarrow C_3$), explain how this preliminary step would affect the Gaussian triangularisation, what precaution should be taken with the sign of the determinant, and check whether the same conclusions are reached regarding the values of $k$ for which the inverse of $A(k)$ exists.