Exercises: Gaussian elimination

In a game of a well-known strategy video game, the statistics of three players (attack points, defence, and speed) have been organized into the following matrix $A$:

$$A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 5 & 3 \\ 1 & 0 & 2 \end{pmatrix}$$

Indicate which elementary row operation you should perform first to obtain a zero in position $a_{21}$ and which for position $a_{31}$, using the first row ($F_1$) as the pivot row. Write down the resulting matrix after applying both changes.

Given the following matrix $B$, which represents the stock levels of two warehouses of a sportswear shop:

$$B = \begin{pmatrix} 1 & -3 \\ 2 & k \end{pmatrix}$$

Determine the value of the real parameter $k$ so that, when applying Gaussian elimination to triangularise the matrix, the second row becomes a row of zeros. What could you say about the rank of the matrix in that case?

A social media analyst studies the interaction between three influencers. The adjacency matrix $M$ shows the direct mentions between them. To simplify the study of the rank of connections, it is required to triangularise the matrix:

$$M = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 1 \\ 1 & 3 & 1 \end{pmatrix}$$

Use Gaussian elimination to transform $M$ into an upper triangular matrix.

When calculating the rank of a matrix, it is fundamental to identify dependent rows. Consider the matrix $C$:

$$C = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 6 & 9 \\ 2 & 4 & 7 \end{pmatrix}$$

Apply Gaussian elimination. How many non-zero rows remain after triangularisation? Justify, from the result, what the rank of the matrix $C$ is.

Find the inverse matrix of the square matrix $A$ of order 2 using the Gauss-Jordan method:

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

An urban logistics company organises the routes of its electric delivery vans across three districts. Matrix $D$ describes the possible routes. To optimise the system, the technical team needs to know the rank of this matrix using Gaussian elimination:

$$D = \begin{pmatrix} 1 & -1 & 2 \\ 2 & -2 & 4 \\ 1 & 2 & 3 \end{pmatrix}$$

Perform the necessary transformations to triangularise the matrix and determine whether any route is redundant (if any row depends on the others).

Given the matrix $S$, perform the steps of Gaussian elimination to obtain an echelon matrix:

$$S = \begin{pmatrix} 2 & 4 & -2 \\ 1 & 3 & 2 \\ -1 & -2 & 1 \end{pmatrix}$$

To simplify calculations, it is recommended to swap the first row ($F_1$) with the second ($F_2$) first, in order to work with a 1 in the pivot position $a_{11}$. Write down the final matrix obtained.

Three servers in a network, $S_1$, $S_2$ and $S_3$, exchange data packets. The dependency relation between their flows is expressed in matrix $K$. To analyse the independence of the flows, calculate the rank of $K$ by applying Gaussian elimination:

$$K = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{pmatrix}$$

Visualise the problem as a dependency graph.

A logistics company uses a matrix to organize the inventory of three regional warehouses ($A_1, A_2, A_3$) with respect to four types of products (electronics, textiles, home goods and food). The stock matrix is as follows:

$$M = \begin{pmatrix} 1 & 2 & -1 & 3 \\ 2 & 3 & 0 & 1 \\ 4 & 7 & -2 & 7 \end{pmatrix}$$

Use Gaussian elimination to determine the rank of the matrix $M$. What does it mean, from an inventory perspective, if the rank is not maximum?

Calculate the inverse matrix of $A$ using the Gauss-Jordan algorithm. The matrix represents a transformation of coordinates in technical design software:

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

Check that the process is correct by verifying that $A \times A^{-1} = I$.

In the study of the stability of a metallic structure, the following matrix depending on a real parameter $k$, which represents the tension at a node, has been obtained:

$$A = \begin{pmatrix} 1 & k & 1 \\ 2 & 1 & k \\ 0 & 1 & 1 \end{pmatrix}$$

Determine, using Gaussian elimination, for which values of $k$ the rank of the matrix is equal to 2.

Given the matrices $A$ and $B$, solve the matrix equation $A \times X = B$ by first finding the inverse of $A$ using Gaussian elimination:

$$A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1 \end{pmatrix} \quad B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{pmatrix}$$

A video game programmer is working on collision detection. To simplify the calculations, they need to know if the direction vectors of three objects are linearly independent. The vectors form the rows of the matrix $D$:

$$D = \begin{pmatrix} 1 & -1 & 3 \\ 2 & 0 & 1 \\ -1 & 3 & -8 \end{pmatrix}$$

Apply Gaussian elimination to triangularise the matrix and determine whether the programmer can simplify their system by removing any redundant vector (that is, if the rank is less than 3).

The following graph shows the server network of a small local network. The associated adjacency matrix indicates the direct connections between nodes $S_1, S_2, S_3$ and $S_4$.

If the adjacency matrix of this graph is $A$, use Gaussian elimination to find the rank of the matrix $A^2$.

$$A = \begin{pmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{pmatrix}$$

Find the inverse matrix of $B$ using the Gauss-Jordan method, knowing that $B$ is a scalar matrix modified by a read error in one of its components:

$$B = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 1 & 1 & 2 \end{pmatrix}$$

Show all intermediate steps of the elementary row operations.

Discuss the rank of the matrix $A$ according to the values of the parameter $m$ using Gaussian elimination:

$$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & m & 6 \\ 3 & 6 & 9 \end{pmatrix}$$

Is there any value of $m$ for which the matrix is invertible? Reason your answer based on the rank obtained.

Determine the rank of the following matrix $A$ depending on the values of the real parameter $k$, using exclusively Gaussian elimination:

$A = \begin{pmatrix} 1 & k & -1 & 1 \\ 2 & 1 & -k & 2 \\ k & 1 & -1 & k \end{pmatrix}$

Analyse in detail the cases where the rank is maximum and the values of $k$ that cause a reduction in the number of linearly independent rows.

Calculate the inverse matrix of $M$ using the Gauss-Jordan method, showing all intermediate steps of the elementary row operations:

$M = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & -1 \\ 2 & 3 & 1 \end{pmatrix}$

Once $M^{-1}$ is obtained, check the result by verifying that $M \times M^{-1} = I$.

In the design of a server network for a new social network, data transfer connections have been established and are represented by the following adjacency matrix $S$. Due to a protocol error, the transfer is only stable if the matrix is invertible.

$S = \begin{pmatrix} 1 & 0 & 2 \\ -1 & k & 1 \\ 0 & 1 & k \end{pmatrix}$

Use Gaussian elimination to find the values of $k$ for which the server network is unstable (not invertible). For the value $k = 2$, calculate the inverse matrix that would represent the data return protocol.

Solve the matrix equation $A \times X \times A = B$, where:

$A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{pmatrix} \quad y \quad B = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}$

To do this, first obtain the inverse of $A$ using Gaussian elimination and subsequently rearrange for $X$ by carrying out the corresponding matrix products.

Consider the square matrix of order 4 depending on the parameter $m$:

$B = \begin{pmatrix} 1 & 1 & 1 & 1 \\ m & 2 & 2 & 2 \\ 1 & m & 3 & 3 \\ 1 & 1 & m & 4 \end{pmatrix}$

Use Gaussian elimination to triangularise the matrix and determine for which values of $m$ the rank of $B$ is less than 4. Is there any value of $m$ for which the rank is 2? Justify your answer following the reduction process.

An encryption system uses a matrix $E$ to encode three-letter messages by transforming them into column vectors. To be able to decrypt, the matrix $D = E^{-1}$ is required. If the encryption matrix is:

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

Calculate the decryption matrix $D$ using the Gauss-Jordan method. Work with fractions to maintain the exact precision required in the cipher.

Given the matrix $A$, calculate its rank according to the values of the real parameters $a$ and $b$:

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

Use Gaussian elimination to transform the matrix into echelon form and discuss all possible combinations of $a$ and $b$ that affect the linear independence of the rows.

It is known that a matrix $X$ satisfies the relation $A \times X = B - 2X$, where $A$ is a square matrix. To be able to rearrange for $X$, the matrix $(A + 2I)$ must be invertible. Given:

$A = \begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & 2 \\ -1 & 0 & 1 \end{pmatrix} \quad y \quad B = \begin{pmatrix} 3 & 0 & 1 \\ 1 & 2 & 0 \\ 0 & 1 & 4 \end{pmatrix}$

1. Show with reasoned arguments that $(A + 2I)$ is invertible.

2. Calculate the matrix $X$ by obtaining the necessary inverse using Gaussian elimination.