The inverse of a matrix using determinants

In a strategy video game, the transformation coordinates of a unit are managed using the matrix $B$:

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

Find the adjugate matrix, $Adj(B)$, and check whether the matrix $B$ has an inverse matrix by first calculating its determinant.

Calculate the inverse matrix $A^{-1}$ of the following matrix using the determinant method:

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

Remember that the formula is $A^{-1} = \frac{1}{|A|} \cdot (Adj(A))^t$.

An engineer is designing an automatic control system. For the system to be stable, the coefficient matrix $M$ must not be singular (its determinant must be non-zero). Given the matrix:

$M = \begin{pmatrix} m & 4 \\ 1 & m \end{pmatrix}$

Determine for which values of the real parameter $m$ the matrix $M$ is singular and, therefore, the system would not have an inverse.

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

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

1. a) Determine, giving reasons, the values of $k$ for which the matrix $A$ does not have an inverse.
2. b) For the case $k = 0$, calculate the inverse matrix $A^{-1}$ using the adjugate method.

Consider the matrix $A = \begin{pmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{pmatrix}$.

1. a) Find the polynomial $P(x) = |A|$.
2. b) Determine the values of $x$ for which $A$ has no inverse.
3. c) For the values of $x$ for which $A$ is invertible, find the expression for $A^{-1}$ in terms of $x$.
4. d) Verify that for $x=0$, $A^{-1} = \frac{1}{2}(A^2 - 3I)$ holds true.

It is known that the determinant of a square matrix $X$ of order 3 is $|X| = -4$. Without calculating the elements of the matrices, find the value of the following determinants:

• a) $|X^{-1}|$

• b) $|X^2|$

• c) $|X \cdot X^t|$

Let $A$ and $B$ be two square matrices of order 3 such that $|A| = 2$ and $|B| = -3$. Using the properties of determinants, calculate the value of the following determinants:

1. a) $|A^t \cdot B|$
2. b) $|(2A)^{-1}|$
3. c) $|A \cdot B^{-1} \cdot A^2|$
4. d) $|B \cdot adj(B)|$ (Recall the relationship between a matrix, its adjugate and its determinant).

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

$A = \begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & -1 \\ 2 & 6 \end{pmatrix}$

To do this, first calculate the inverse of $A$ using determinants and rearrange for the matrix $X$.

Solve the following matrix equation $X \cdot A + B = C$, where:

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

First justify why it is possible to find the inverse matrix of $A$ and rearrange for the unknown $X$ before performing the numerical calculations.