Inverse matrix

Given the matrix of coefficients of a cost system in a small café:

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

Determine for which value of the parameter $k$ the matrix does not have an inverse (that is, its determinant is zero).

In an eSports tournament, the efficiency of two teams is measured by comparing their results on two different maps. The data are organized in the following matrix $M$, where each row corresponds to a team (A and B) and each column to a map (1 and 2):

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

Calculate the determinant of the matrix $M$ to obtain the tournament's competitiveness index.

To prepare for an algebra exam, a student needs to calculate the adjugate matrix of the following $2 \times 2$ matrix. Remember that $Adj(A)$ is the transpose of the cofactor matrix of $A$, where each cofactor is the sign $(-1)^{i+j}$ multiplied by the minor of the corresponding element. Find $Adj(A)$ given:

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

The figure shows the matrix $A$ and the sign pattern $(-1)^{i+j}$ for a $2 \times 2$ matrix.

Given the matrix $A$ which depends on a real parameter $x$:

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

Work out:

1. Calculate the determinant of matrix $A$ in terms of $x$.

2. Determine for which values of $x$ the matrix does not have an inverse.

3. For the case $x = 0$, justify whether the inverse matrix exists and, if so, calculate only the element $a_{21}$ of the matrix $A^{-1}$ (the element in the second row and first column).

Calculate the inverse matrix $A^{-1}$ of the matrix $A$ using the formula $A^{-1} = \frac{1}{|A|} \cdot (Adj(A))^t$.

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

Note: Do not forget to check first that the determinant is non-zero.

We know that the determinant of a square matrix $M$ of order 3 is $|M| = -2$. Using the properties of determinants, calculate, giving reasons, the value of the following determinants:

1. $|M^t \cdot M|$, where $M^t$ is the transpose matrix of $M$.

2. $|3M|$.

3. The determinant of the inverse matrix, $|M^{-1}|$.

4. The determinant of a matrix $P$ obtained by swapping the first and third rows of $M$ and then multiplying the second column by $-5$.

Solve the following matrix equation by rearranging for the unknown $X$. It is given that $A \cdot X = B$, where:

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

Remember that to solve for $X$ you must multiply on the left by the inverse of $A$.

Given the matrix $M$ depending on the real parameter $k$:

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

1. a) Determine for which values of $k$ the matrix $M$ does not have an inverse matrix.
2. b) For $k = 0$, solve the matrix equation $M \cdot X + M = I$, where $I$ is the identity matrix of order 3.