Exercises: Laplace expansion

Given the following $3 \times 3$ matrix $A$, identify and calculate only the minor of the element $a_{21}$ (the number $4$) and its corresponding cofactor $A_{21}$, paying special attention to the sign.

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

The figure highlights the row and the column that the element $a_{21}$ belongs to.

Calculate the value of the determinant of the matrix $B$ by expanding along the second row. Explain why choosing this row makes the calculation easier compared to the first row.

$$|B| = \begin{vmatrix} 5 & 7 & 1 \\ 0 & 2 & 0 \\ 3 & -1 & 4 \end{vmatrix}$$

To understand Laplace's method, it is essential to know the "chessboard rule" for the signs of cofactors. Complete the sign matrix for a matrix of order $4 \times 4$ and determine which sign corresponds to the cofactor of element $a_{32}$.

A marketing analyst is studying the interaction between three social networks in an advertising campaign. The impact matrix is as follows:

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

Calculate the determinant of $M$ by expanding along the first column.

Calculate the determinant of the following matrix of order 4. Although it may seem complex due to its size, use Laplace's method by selecting the row or column containing the most zeros to simplify the process.

$$|D| = \begin{vmatrix} 1 & 2 & 0 & 3 \\ 0 & 0 & 5 & 0 \\ 2 & 1 & -1 & 1 \\ 4 & 3 & 0 & 2 \end{vmatrix}$$

Find the value of the parameter $k$ so that the determinant of the following matrix is equal to zero, using expansion along the elements of the third row.

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

Given matrix $C$, calculate its determinant by expanding along the second column. Then check the result by applying Sarrus' rule to verify that both methods match.

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

In a strategy video game, the resources of three villages are distributed according to matrix $R$. If the determinant of this matrix represents the "Equilibrium Factor", calculate it by expanding along the first row.

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

To calculate the determinant of the matrix $B = \begin{pmatrix} 2 & -3 & 1 \\ 4 & 0 & -2 \\ 1 & 5 & 3 \end{pmatrix}$, Lucía decides to expand using Laplace's expansion along the second row to take advantage of the zero. She writes on her exam:

$$|B| = 4 \cdot \begin{vmatrix} -3 & 1 \\ 5 & 3 \end{vmatrix} + 0 \cdot A_{22} - 2 \cdot \begin{vmatrix} 2 & -3 \\ 1 & 5 \end{vmatrix}$$

Since $\begin{vmatrix} -3 & 1 \\ 5 & 3 \end{vmatrix} = -9 - 5 = -14$ and $\begin{vmatrix} 2 & -3 \\ 1 & 5 \end{vmatrix} = 10 - (-3) = 13$, she concludes that $|B| = 4 \cdot (-14) - 2 \cdot 13 = -56 - 26 = -82$.

1. a) Identify the conceptual error Lucía made when applying the definition of cofactor ($A_{ij} = (-1)^{i+j} \cdot M_{ij}$) in her expansion.
2. b) Explain how the position $(i, j)$ determines the sign that accompanies each minor according to the checkerboard pattern rule.
3. c) Correct the approach and calculate the correct value of $|B|$.

Consider the following square matrix of order 4:

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

To calculate its determinant, three different methods have been proposed:

• Method 1: Directly apply Sarrus' rule by multiplying the diagonals of four elements.

• Method 2: Apply Laplace's expansion along the elements of the first row.

• Method 3: Apply Laplace's expansion along the elements of the second row.

1. a) Reason why Method 1 is conceptually incorrect and cannot be applied to this matrix.
2. b) Compare Methods 2 and 3: calculate how many order 3 determinants each would require solving and justify which of the two is the most efficient.
3. c) Following the most efficient method you have chosen, calculate the value of $|A|$ clearly showing the cofactor used.

Calculate the value of the following determinant of order 4 using expansion by the elements of the row or column you consider most appropriate (Laplace expansion):

$|A| = \begin{vmatrix} 2 & -1 & 0 & 3 \\ 1 & 0 & 4 & 0 \\ 3 & 1 & 2 & 1 \\ 0 & 5 & 0 & -2 \end{vmatrix}$

Given the matrix $M$, calculate its determinant by expanding along the second row. Subsequently, check whether the result matches by expanding along the third column:

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

Use this exercise to verify that the value of the determinant is independent of the row or column chosen for cofactor expansion.

In calculating determinants using Laplace expansion, the sign of each term depends on the position of the element. For the following "light" matrix of a billboard panel (where each number represents intensity), calculate the determinant by expanding along the column containing the most zero values:

$|A| = \begin{vmatrix} 1 & 0 & 4 \\ 3 & 2 & 0 \\ 5 & 0 & 1 \end{vmatrix}$

Find the values of the parameter $k$ for which the determinant of the following matrix is zero, using cofactor expansion along the first row:

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

A data analyst processes an adjacency matrix of a social network. To verify the stability of the network, they need to calculate the determinant of the matrix $B$. First apply properties of determinants to make as many zeros as possible in the first column, and then expand using Laplace expansion:

$|B| = \begin{vmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 1 & 4 & 9 & 16 \\ 1 & 1 & 5 & 2 \end{vmatrix}$

Solve the following equation knowing that the determinant has been calculated by expanding along the third row:

$\begin{vmatrix} x & 2 & 3 \\ 1 & x & 0 \\ 0 & 1 & -1 \end{vmatrix} = 5$

Given the matrix $A$, calculate its cofactor matrix $Adj(A)$ and use these values to calculate the determinant of $A$ using Laplace expansion, choosing whichever row you prefer:

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

Note: Remember that $|A| = a_{i1}A_{i1} + a_{i2}A_{i2} + a_{i3}A_{i3}$ for any row $i$.

In an economic exchange model, the following matrix of coefficients is presented. Calculate its determinant by combining Sarrus' rule for the resulting minors and Laplace expansion using the column containing the most zeros:

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

Before diving into detailed calculations, analyze the structure of matrix $M$:

$$M = \begin{pmatrix} 20 & 1 & 0 & -2 \\ 0 & 3 & 1 & 0 \\ 0 & 0 & 2 & -1 \\ 1 & 0 & 0 & 2 \end{pmatrix}$$

1. a) Estimate: If we disregarded the entry $a_{41} = 1$, the resulting matrix would be upper triangular. Mentally calculate the product of the elements on its main diagonal and, based on that value, reasonably estimate the sign and the approximate hundred in which the actual determinant $|M|$ should lie.
2. b) Calculate: Expand the determinant of $M$ using Laplace's expansion along the first column (taking advantage of its two zeros) and obtain its exact value.
3. c) Check: Compare the exact result obtained with your previous estimate, find the absolute numerical difference between the two, and conceptually explain why the presence of that off-diagonal entry $a_{41} = 1$ perturbs the final result so little.

Design a square matrix $A$ of order $4 \times 4$ that simultaneously satisfies the following four conditions:

1. It contains exactly 6 zeros in total among its 16 entries.

2. When expanding the determinant $|A|$ using Laplace's expansion along the third column, the expansion reduces to a single non-zero term (that is, it only requires computing one order 3 cofactor).

3. The $3 \times 3$ submatrix resulting from removing the row and column corresponding to that non-zero entry is upper triangular, with all its main diagonal entries non-zero.

4. The final determinant of the matrix is exactly $|A| = -24$.

To solve it:

1. a) Deduce with justification in which positions of the third column and of the rest of the matrix the 6 zeros must necessarily be placed to satisfy conditions 2 and 3.
2. b) Assign coherent numerical values to the non-zero entries taking into account the sign of the cofactor $(-1)^{i+j}$ and the property of the determinant of a triangular matrix, so that the result is $-24$.
3. c) Write your complete matrix $A$ and verify in detail that the Laplace expansion along the third column yields the required value.

In a market study on the competition between four streaming platforms (Netflix, HBO, Disney+, and Amazon Prime), the monthly transfer of users has been modeled using matrix $M$. For the system to be stable and not collapse financially, the determinant of the transition matrix must be zero.

Given the matrix:

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

Calculate, using Laplace's expansion, the values of $x$ that make $|M| = 0$. Hint: Strategically choose the row or column to expand the determinant or apply prior properties to generate additional zeros.

A logistics company organizes its routes between four main nodes. Matrix $A$ represents the freight flow connections. It is known that the efficiency of the network is related to the value of the determinant of said matrix.

Calculate the determinant of matrix $A$ by cofactor expansion along the second column, after having performed at least one elementary row or column operation to simplify the calculation:

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

Let $A$ be a square matrix of order 3 whose determinant is $|A| = 5$. Consider the matrix $B$ of order 4 defined as follows:

$$B = \begin{pmatrix} 1 & 2 & 0 & 5 \\ 0 & a_{11} & a_{12} & a_{13} \\ 0 & a_{21} & a_{22} & a_{23} \\ 0 & a_{31} & a_{32} & a_{33} \end{pmatrix}$$

Where the elements $a_{ij}$ correspond to the elements of the original matrix $A$.

1. Justify with reasoning, by applying Laplace expansion along the most appropriate row or column, what the value of the determinant of $B$ is.

2. If we multiply the first row of $B$ by $3$ and add to it the second row multiplied by $2$, how does this affect the value of the determinant calculated previously?

In the analysis of an index investment portfolio, a matrix of partial correlation coefficients is used. To determine whether the matrix is invertible (and therefore the portfolio is diversifiable), its determinant must be calculated.

Find the value of the determinant of the following matrix using expansion by the elements of a row or column (Laplace expansion), after first seeking a linear combination of rows or columns that generates the maximum number of zeros in the third row:

$$C = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 2 & 3 & 4 & 5 \\ 4 & 9 & 16 & 25 \\ 1 & 2 & 4 & 8 \end{pmatrix}$$

Consider the parametric matrix $P$ associated with a production cost model:

$$P = \begin{pmatrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{pmatrix}$$

Calculate the determinant of the matrix $M$ of order 4 using Laplace expansion, where $M$ is constructed by block partitioning the matrix $P$ as follows:

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

Determine for which values of the parameter $k$ the determinant of $M$ is equal to $k^2$.

Given the matrix $A = (a_{ij})$ of order $3 \times 3$ such that its determinant is $|A| = k$. A new matrix $H$ of order 4 is constructed by introducing a "control" row at the top and an "identity" column on the left:

$$H = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & a_{11} & a_{12} & a_{13} \\ 0 & a_{21} & a_{22} & a_{23} \\ 0 & a_{31} & a_{32} & a_{33} \end{pmatrix}$$

1. Show, by expanding along the first column using Laplace expansion, that $|H| = |A|$.

2. If the elements of the matrix $A$ are such that $a_{ij} = i + j$, calculate the exact value of $|H|$ by applying Laplace expansion along the row or column you consider most advantageous.

An encryption algorithm uses square matrices of order 4. To decrypt a message, it is vital that the determinant of the key matrix $K$ is non-zero.

Use Laplace expansion to calculate the determinant of $K$. Note: Before expanding by cofactors, use the properties of determinants to make all the elements in the fourth column, except one, equal to zero.

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

Let $A$ be the matrix of order 4 defined by:

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

Determine the values of $x$ for which the matrix $A$ has an inverse.

A logistics company based in Zaragoza models the transfer of goods between four key regional nodes: Madrid ($N_1$), Zaragoza ($N_2$), Barcelona ($N_3$), and Valencia ($N_4$). The dynamic flow matrix $T(k)$ depends on a winter congestion index $k \in \mathbb{R}$:

$$T(k) = \begin{pmatrix} 1 & k & 0 & 1 \\ 2 & 1 & -1 & 0 \\ 0 & 3 & k & 2 \\ 1 & 0 & 0 & -1 \end{pmatrix}$$

Network theory states that the system experiences a "functional deadlock" (loss of linear independence among its routes) if the determinant of the matrix vanishes, that is, when $|T(k)| = 0$.

1. a) Apply Laplace's expansion along the row or column containing the greatest number of zeros to obtain the simplified polynomial expression of $P(k) = |T(k)|$, detailing the calculation of each order 3 minor used.
2. b) Algebraically determine all real values of the congestion index $k$ for which the transport system enters a functional deadlock.
3. c) A feasibility report from the Directorate-General for Traffic warns that during a cold snap the actual index $k$ always fluctuates within the interval $[1{,}5 \,;\, 2{,}0]$. Conclude with reasoning whether the network is expected to collapse due to a functional deadlock under those weather conditions.

To compute the determinant of the parametric matrix of order 4 ($a, b, c \in \mathbb{R}$, with $a \cdot b \cdot c \neq 0$):

$$M = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1+a & 1 & 1 \\ 1 & 1 & 1+b & 1 \\ 1 & 1 & 1 & 1+c \end{pmatrix}$$

three 2nd-year Bachillerato students propose different approaches:

• Marcos suggests expanding directly using Laplace's expansion along the first row.

• Sara proposes first transforming the matrix by subtracting the first row from the others ($F_2 \to F_2 - F_1$, $F_3 \to F_3 - F_1$, $F_4 \to F_4 - F_1$) and then applying Laplace.

• Iván claims that it is faster to perform two cross operations simultaneously: $F_1 \to F_1 - F_2$ and $F_2 \to F_2 - F_1$.

1. a) Evaluate Iván's proposal: explain which mathematical property of elementary transformations is violated when performing simultaneous circular operations and what error it would cause in the determinant.
2. b) Compare the computational workload of Marcos's option versus Sara's: specify how many $3 \times 3$ determinants with parameters Marcos would have to calculate and what structure the matrix acquires with Sara's proposal.
3. c) Justify why the row operations proposed by Sara do not alter the value of $|M|$, carry them out, and expand using Laplace along the most suitable line to obtain the final factored formula for the determinant in terms of $a$, $b$, and $c$.