Laplace's method

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.

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 $a_{32}$.

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.

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

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.

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):

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

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.

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

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.

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.