Exercises: 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 check the consistency of the inventory data.

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 of order 2:

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

Calculate, giving reasons and 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 $|2A|$.

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 $A^{-1} = \frac{1}{|A|} \times (Adj(A))^t$.

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) $|X \times X^t|$

Solve the matrix equation $A \times X = B$, 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 $X$.

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:

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

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

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.

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

Solve the following matrix equation $X \times A + B = C$, 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.

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.

1. 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?

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 $M \times X = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$.

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 using Sarrus' rule, the value of the determinant of the following matrix $B$:

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

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

A square matrix $A$ is said to be orthogonal if it satisfies the condition $A \times A^t = I$, 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 whether 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 $\alpha$.

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 $K^{-1}$.

Determine the values of the real parameter $a$ for which the following determinant of order 4 is equal to zero. To solve it, use only 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 this structure is similar to a Vandermonde determinant).

A robotics company is designing the control system for a mechanical arm. The relationship between the sensors and actuators is given by the matrix equation $A \times X \times A = B$, 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) Find the values of $k \in \mathbb{R}$ for which the matrix $A$ is non-singular (invertible).
2. b) For $k = 2$, rearrange the equation for the matrix $X$ and calculate it using the adjugate matrix method to find the required inverses.

In the development of a new spaceship video game, the coordinates of objects in 3D space are transformed using 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 the position $(1, 1, 1)$ after applying $M$ with $\alpha = \frac{\pi}{2}$ and $a = 0$, what were its original coordinates?

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 find, with reasoning, the value of:

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

An encryption system uses a matrix $C$ to transform blocks of messages. For the message to be recoverable, the rank of the matrix must be maximum. Determine 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.

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.

A square matrix $A$ is known to satisfy the following equality: $A^2 - 5A + 6I = O$, where $I$ is the identity matrix and $O$ is the zero matrix.

1. a) Prove 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 $A = \begin{pmatrix} 4 & 2 \\ -1 & 1 \end{pmatrix}$, verify that it satisfies the initial equation and find its inverse using the result from section (b).

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 factorise 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 $A \times \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ using the inverse matrix found in the previous part.