Inverse of a Matrix Using Determinants

In a small logistics company, the number of pallets of three types of products in three different warehouses is recorded using the following matrix $A$:

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

Calculate the determinant of the matrix $A$ using Sarrus' rule to verify the consistency of the inventory data.

Given the matrix of order 2:

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

Calculate with reasoning, applying the properties of determinants:

• a) The value of $|A|$.

• b) The value of the determinant of its transpose, $|A^t|$.

• c) The value of $|2A|$.

Given that the determinant of a matrix $A = \begin{pmatrix} x & y & z \\ p & q & r \\ a & b & c \end{pmatrix}$ is equal to $5$, use the properties of determinants to calculate, without expanding by Sarrus' rule, the value of the determinant of the following matrix $B$:

$$B = \begin{pmatrix} 2x & 2y & 2z \\ p+x & q+y & r+z \\ 3a & 3b & 3c \end{pmatrix}$$

Indicate in detail which property you apply in each step of the simplification.

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.

Determine the values of the real parameter $a$ for which the following determinant of order 4 is zero. To solve it, use exclusively the properties of determinants to simplify the expression before expanding along a row or column:

$$ |A| = \begin{vmatrix} 1 & 1 & 1 & 1 \\ a & 2 & -1 & 3 \\ a^2 & 4 & 1 & 9 \\ a^3 & 8 & -1 & 27 \end{vmatrix} $$

(Hint: Notice that it has a structure similar to a Vandermonde determinant).

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

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

Find the adjugate matrix, $Adj(B)$, and check if 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 $A^{-1} = \frac{1}{|A|} \cdot (Adj(A))^t$.

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

A basic encryption system uses a matrix $K$ to transform messages in vector form. For the message to be decrypted, the matrix $K$ must be invertible. Given the matrix:

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

1. a) Find the values of $m$ that make the encryption system insecure (the matrix is not invertible).
2. b) If $m = 1$, calculate the decryption matrix $K^{-1}$.

It is known that the determinant of a square matrix $X$ of order 3 is $|X| = -4$. Without calculating the elements of the matrices, obtain 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) $|B \cdot adj(B)|$ (Remember the relationship between a matrix, its adjugate and its determinant).

Let $A$ be a square matrix of order 3 such that its determinant is $|A| = \frac{1}{2}$. Without explicitly calculating the elements of the matrix, use the properties of determinants to reasonably find the value of:

1. a) $|A^{-1} \cdot (2A)^t|$
2. b) $|Adj(A)|$, where $Adj(A)$ is the matrix formed by the cofactors of the elements of $A$.
3. c) The determinant of the matrix $B$ if it is known that $A \cdot B = Adj(A)$.

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 isolate 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 isolate the unknown $X$ before performing the numerical calculations.

A robotics company is designing the control system for a mechanical arm. The relationship between the sensors and the actuators is given by the matrix equation $A \cdot X \cdot A = B$, where $A$ is a matrix that depends on the sensitivity $k$ of the sensor:

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

1. a) Investigate for which values of $k \in \mathbb{R}$ the matrix $A$ is non-singular (invertible).
2. b) For $k = 2$, isolate the matrix $X$ in the equation and calculate it using the adjugate matrix method to find the necessary inverses.

In the design of a digital filter for audio signals, a transfer matrix $H$ is used, defined as:

$$H(x) = \begin{pmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{pmatrix}$$

1. a) Find the values of the real parameter $x$ for which the determinant of the matrix is zero.
2. b) For $x = 2$, calculate the inverse matrix $H^{-1}$. What is the significance of the determinant being very close to zero in engineering applications?

In the development of a new spaceship video game, the coordinates of objects in 3D space are transformed by a rotation and scaling matrix $M$. For the graphics engine to be able to "undo" the transformation, the matrix must have an inverse.

$$M = \begin{pmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & e^a \end{pmatrix}$$

1. a) Show that the determinant of $M$ is $|M| = e^a$ for any value of the angle $\alpha$.
2. b) Calculate the inverse matrix $M^{-1}$ in terms of $\alpha$ and $a$ using determinants.
3. c) If an object ends up at position $(1, 1, 1)$ after applying $M$ with $\alpha = \frac{\pi}{2}$ and $a = 0$, what were its original coordinates?

Given the matrix $A = \begin{pmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{pmatrix}$:

1. a) Calculate $|A|$ and factorize the result.
2. b) If $a=1, b=2, c=3$, calculate the inverse matrix $A^{-1}$ using the adjugate method.
3. c) Solve the system of equations $A \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ using the inverse matrix found in part (b).

Consider the matrix $M = \begin{pmatrix} a & 0 & 2 \\ a & a & 1 \\ 0 & a & -1 \end{pmatrix}$.

1. a) Calculate $|M|$ in terms of $a$.
2. b) Determine the rank of the matrix $M$ according to the values of $a$.
3. c) For $a = 1$, solve the matrix equation $M \cdot X = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$.

An encryption system uses a matrix $C$ to transform message blocks. For the message to be recoverable, the rank of the matrix must be maximum. Investigate the rank of the following matrix in terms of the real parameters $m$ and $n$:

$$C = \begin{pmatrix} 1 & m & 0 & 1 \\ n & 1 & 1 & 0 \\ 1 & 1 & 0 & m \end{pmatrix}$$

Determine for which values of $m$ and $n$ the rank of $C$ is equal to 2.

One of the most interesting applications of determinants in geometry is calculating areas. The area of a triangle whose vertices are $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$ is given by the formula:

$Area = \frac{1}{2} \cdot | \det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} |$

Calculate the area of the triangle represented in the following graph whose vertices are $P(1, 1)$, $Q(5, 2)$ and $R(2, 4)$.

A square matrix $A$ is said to be orthogonal if it satisfies the condition $A \cdot A^t = I$, where $A^t$ is the transpose and $I$ is the identity matrix.

1. a) Show that if $A$ is orthogonal, then its determinant can only be $1$ or $-1$.
2. b) Check if the matrix $A = \begin{pmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{pmatrix}$ is orthogonal and find its inverse $A^{-1}$ in terms of $\alpha$.

It is known that a square matrix $A$ satisfies the following equation: $A^2 - 5A + 6I = O$, where $I$ is the identity matrix and $O$ is the zero matrix.

1. a) Show with reasoning that the matrix $A$ is invertible.
2. b) Express the inverse matrix $A^{-1}$ as a linear combination of the matrices $A$ and $I$.
3. c) If $A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}$, verify that it satisfies the initial equation and find its inverse using the result from part (b).

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 $A$ does not have an 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.