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

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) $|X \cdot X^t|$

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 $X$.

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

Hugo needs to solve the matrix equation $A \cdot X + B = C$, where:

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

On his exam, Hugo writes the following steps:

1. He rearranges the equation as $A \cdot X = C - B = \begin{pmatrix} 3 & 3 \\ -1 & 3 \end{pmatrix}$.

2. He finds $|A| = 1 \cdot 3 - 2 \cdot 1 = 1 \ne 0$, deducing that the inverse matrix $A^{-1}$ exists.

3. He calculates the cofactors of $A$: $A_{11}=3$, $A_{12}=-1$, $A_{21}=-2$, $A_{22}=1$, and directly writes $\operatorname{Adj}(A) = \begin{pmatrix} 3 & -1 \\ -2 & 1 \end{pmatrix}$.

4. He claims that $A^{-1} = \dfrac{1}{|A|} \cdot \operatorname{Adj}(A) = \begin{pmatrix} 3 & -1 \\ -2 & 1 \end{pmatrix}$.

5. Finally, he calculates the solution as $X = (C - B) \cdot A^{-1} = \begin{pmatrix} 3 & 3 \\ -1 & 3 \end{pmatrix} \begin{pmatrix} 3 & -1 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ -9 & 4 \end{pmatrix}$.

Carry out the following tasks:

1. a) Identify and explain, with reasoning, the two conceptual errors that Hugo made (one in obtaining $A^{-1}$ and the other in isolating $X$).
2. b) Correct both mistakes: calculate the correct inverse matrix $A^{-1}$ using the cofactor formula, and determine the matrix $X$ that satisfies the original equation.

Given the square matrix of order $3$:

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

1. a) Before calculating any adjugate (cofactor) or inverse matrix, deduce with reasoning the sign and exact value of $|A^{-1}|$ and of $|\operatorname{Adj}(A)|$ using exclusively the determinant $|A|$ and the theoretical properties of determinants.
2. b) Calculate step by step the 9 cofactors of $A$, construct the cofactor matrix $\operatorname{Adj}(A)$, and obtain the inverse matrix $A^{-1}$ applying the formula $A^{-1} = \dfrac{1}{|A|} \cdot (\operatorname{Adj}(A))^t$.
3. c) Check the consistency of your deductions by directly calculating the determinant of the matrix $\operatorname{Adj}(A)$ obtained in part b), and verifying that the matrix product $A \cdot A^{-1}$ yields the identity matrix $I_3$.

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) $|A^t \cdot B|$
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 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.

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

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 $A \cdot A^t = I$, 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 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 $K^{-1}$.

Consider the matrix of order $3$ depending on the real parameter $k$:

$$M = \begin{pmatrix} 1 & 2 & -1 \\ 0 & k & 1 \\ 2 & 1 & 0 \end{pmatrix}$$

Two independent tasks are proposed:

1. Determine for which values of $k$ the matrix $M$ is non-singular (has an inverse).

2. For the value $k = 1$, calculate only the entry in row 3 and column 2 of the inverse matrix, that is, $(M^{-1})_{32}$.

1. a) For task 1, compare the strategy of row-reducing the matrix using Gaussian elimination versus calculating its determinant $|M|$. Justify which of the two approaches is more convenient in this case and use it to give the answer.
2. b) For task 2, one alternative is to calculate the full inverse matrix $M^{-1}$ and then select that entry; another alternative is to use the formula $M^{-1} = \dfrac{1}{|M|} \cdot (\operatorname{Adj}(M))^t$ to calculate only the necessary cofactor. Justify which of the two paths requires less computational effort, deduce precisely which cofactor $M_{ij}$ is involved (justifying the role of the transpose), and calculate the value of $(M^{-1})_{32}$.

A technology consulting firm based in Seville sends its three audit teams by high-speed train (Renfe AVE) during a quarter, purchasing tickets in three different fares: Premium ($x$ €), Elige ($y$ €), and Básico ($z$ €). The accounting records for the quarter show the following purchase data:

• In October they purchased 1 Premium ticket, 1 Elige, and 1 Básico, with a total expenditure of $180$ €.

• In November they purchased 2 Premium tickets, 1 Elige, and 2 Básico tickets, totaling an expenditure of $300$ €.

• In December they purchased 1 Premium ticket, 3 Elige, and 2 Básico tickets, with a total cost of $330$ €.

1. a) Model the situation using a matrix equation of the form $A \cdot X = B$, clearly indicating the matrices $A$, $X$, and $B$ along with their dimensions.
2. b) Prove that the coefficient matrix $A$ is invertible by calculating its determinant $|A|$, and find its inverse matrix $A^{-1}$ applying the adjugate (transpose of the cofactor matrix) method.
3. c) Algebraically isolate the unknown vector $X$, justifying the order of matrix multiplication, compute the product $A^{-1} \cdot B$, and determine the exact unit price of each of the three fares.

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 $A \cdot X \cdot A = B$, 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) 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) $|A^{-1} \cdot (2A)^t|$
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 $A \cdot B = Adj(A)$.

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

A square matrix $A$ of order $3$ with real coefficients has as its cofactor matrix:

$$\operatorname{Adj}(A) = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}$$

1. a) Knowing that for any invertible matrix of order $n$, $|\operatorname{Adj}(A)| = |A|^{n-1}$ holds, deduce the possible values of the determinant $|A|$ without previously calculating the elements of matrix $A$.
2. b) From the fundamental identity $A \cdot (\operatorname{Adj}(A))^t = |A| \cdot I_3$, algebraically isolate matrix $A$ in terms of $|A|$ and the inverse matrix of $(\operatorname{Adj}(A))^t$.
3. c) Explicitly determine all matrices $A$ compatible with the given cofactor matrix and algebraically justify why there are exactly two distinct solutions.

Let $A$ be an invertible square matrix of order $3$ whose determinant is $|A| = 2$.

1. a) From the relation $|\operatorname{Adj}(M)| = |M|^{n-1}$ and the property of the determinant of the inverse, formally prove that $|\operatorname{Adj}(A^{-1})| = \dfrac{1}{|\operatorname{Adj}(A)|}$ and calculate its exact numerical value for matrix $A$.
2. b) The matrix $B = 2 \cdot A^t \cdot A^{-1}$ is defined. Justify with reasoning whether matrix $B$ is invertible by first calculating the value of its determinant $|B|$.
3. c) Consider the matrix equation $(\operatorname{Adj}(A))^t \cdot X = A^2$, where $X$ is an unknown square matrix of order $3$. Using the property $A \cdot (\operatorname{Adj}(A))^t = |A| \cdot I_3$, isolate matrix $X$ in terms of $A$ and show that the determinant of $X$ is independent of the numerical elements forming matrix $A$, calculating its exact value.