Exercises: Inverse matrix

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.

A courier company uses three types of vans to deliver parcels to three areas of the city. The number of parcels delivered per day is summarised in matrix $A$. Calculate the determinant of $A$ using Sarrus' rule.

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

Examine the following matrices without performing extensive calculations. State, justifying your answer using the properties of determinants, which of them have a determinant equal to zero:

1. a) $B = \begin{pmatrix} 1 & 5 \\ 2 & 10 \end{pmatrix}$

1. b) $C = \begin{pmatrix} 3 & 4 & 1 \\ 0 & 0 & 0 \\ 1 & 2 & 7 \end{pmatrix}$

1. c) $D = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \end{pmatrix}$

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).

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.

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.

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 $A$.

In a basic cryptography game, matrix $C$ was used to encode a message of two numbers. To decode it, we need to find the inverse matrix $C^{-1}$.

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

Find $C^{-1}$ and verify the result by evaluating the product $C \cdot C^{-1}$, which must yield the identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

When solving the matrix equation $X \cdot A = B$, with $A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 0 \\ 1 & -2 \end{pmatrix}$, Carlos writes on his exam:

"Since $|A| = 2 \cdot 3 - 1 \cdot 5 = 1 \ne 0$, matrix $A$ can be moved dividing to the other side. Therefore, $X = A^{-1} \cdot B$".

Carry out the following tasks:

1. a) Identify the two conceptual errors made by Carlos: one concerning language and formal valid matrix operations (the misconception of "dividing matrices") and another concerning the order in matrix multiplication.
2. b) Write down the rigorous algebraic isolation of matrix $X$, indicating by which side of each member you must multiply by the inverse matrix.
3. c) Calculate the inverse matrix $A^{-1}$ using the adjugate formula and obtain the solution matrix $X$.

A non-singular matrix $A$ of order 2 has unknown entries, but its inverse matrix is known precisely:

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

1. a) Without previously finding the individual entries of $A$, deduce with reasoning the numerical value of its determinant $|A|$ by applying the properties of determinants.
2. b) Using the property $(A^{-1})^{-1} = A$, reconstruct the entries of the original matrix $A$ by using the adjugate matrix method applied to $A^{-1}$.
3. c) Verify that the determinant of matrix $A$ that you obtained in part b matches exactly the value deduced in part a.

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

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).

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 $-5$.

A mobile app development company uses a "server costs" matrix $C$ to organise its monthly expenses in three regions (Europe, Asia and America). The matrix is given by:

To adjust the budgets for the next quarter, they need to solve the matrix equation $C \cdot X = B$, where $B$ is the gross profits matrix:

Calculate matrix $X$ using the inverse matrix method.

In a block cipher system, matrix $K$ is used as a security key to encode messages of three numbers:

In order for the recipient to decode the message, they need to obtain the inverse matrix $K^{-1}$.

1. Show that matrix $K$ is non-singular (invertible).

2. Calculate step by step the inverse matrix $K^{-1}$ using the cofactor method.

Find the values of the parameter $a$ for which matrix $B$ does not have an inverse:

For the value $a = 1$, solve the matrix equation $X \cdot B = I$, where $I$ is the identity matrix of order 3.

Given matrices $A$ and $B$:

Solve the matrix equation $A \cdot X + A = B$. To do this:

1. Rearrange for matrix $X$ in the equation (giving the form $X = \dots$).

2. Calculate the determinant of $A$ to ensure that you can perform the operation.

3. Obtain the final matrix $X$.

Consider the square matrix $A$ of order 2:

1. a) Verify that the equality $A^2 - I = O$ holds, where $I$ is the identity matrix and $O$ is the zero matrix (both of order 2).

1. b) Use the previous result to express the inverse matrix $A^{-1}$ in terms of matrix $A$ itself.

1. c) Calculate $A^{-1}$ using the usual method (with the determinant and the adjugate matrix) and verify that the result matches the one obtained in the previous part.

Three friends, Alba, Bruno and Carlos, go to a café. The first time, Alba has 2 coffees and 1 soft drink; Bruno has 1 coffee and 2 soft drinks; and Carlos has 3 coffees. The total bill per person can be represented by a consumption matrix $A$ and a price matrix $P$.

If the unit price matrix of the items (coffee and soft drink) is unknown $X = \begin{pmatrix} p_c \\ p_r \end{pmatrix}$ and we know that Alba's total spend was $4,50€$ and Bruno's was $5,40€$:

1. Write the system of equations in matrix form $A \cdot X = B$.

2. Calculate the determinant of the matrix of coefficients $A$.

3. Find the inverse matrix $A^{-1}$ and use it to calculate the price of a coffee and a soft drink.

To solve the matrix equation $A \cdot X = B$, where $A = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \\ 1 & 1 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 4 \\ 2 \\ 9 \end{pmatrix}$, two students propose different strategies:

• Lucía proposes calculating the inverse matrix $A^{-1}$ using the adjugate matrix and then performing the matrix product $X = A^{-1} \cdot B$.

• Marta argues that it is preferable to set up a system of three linear equations in three unknowns with $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and solve it using Gaussian elimination.

1. a) Compare both paths and justify which of the two requires fewer arithmetic operations if the only goal is to solve this specific equation.
2. b) Explain in what economic or business context (for example, in a market study where the technical matrix $A$ remains fixed but the demand vector $B$ changes periodically for 50 different scenarios) Lucía's method would be indisputably more efficient than Marta's.
3. c) Find the inverse matrix $A^{-1}$ using the adjugate formula and solve the given equation by calculating the vector $X$.

An incoming travel agency in Andalusia designs three promotional packages that combine three services: hotel nights ($x$), guided monument visits ($y$), and high-speed train tickets ($z$). The number of services contracted by three tourist groups and the total amount billed in euros to each group is described by the matrix relation $M \cdot P = T$, where:

$M = \begin{pmatrix} 3 & 2 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 1 \end{pmatrix}$, $P = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and $T = \begin{pmatrix} 370 \\ 270 \\ 150 \end{pmatrix}$

1. a) Justify that the price structure is uniquely determined by calculating the determinant of the coefficient matrix $M$.
2. b) Calculate the inverse matrix $M^{-1}$ using adjugates and transposition.
3. c) Solve for the vector $P$ through the explicit use of $M^{-1}$, calculate its numerical value, and interpret in the context of the situation what unit price each hotel night, each guided visit, and each train ticket has.

Given that the determinant of a square matrix $A$ of order 3 is $|A| = -2$, use exclusively the properties of determinants to calculate, giving reasons, the value of:

$| (3A)^{-1} \cdot A^t \cdot (-A) |$

To visualise the concept of the sign rule for cofactors, remember the "chessboard" pattern:

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.

A logistics company uses a security code based on matrices. The key matrix is $A = \begin{pmatrix} 1 & 0 & -1 \\ 2 & 1 & 0 \\ 3 & 0 & 1 \end{pmatrix}$. To decrypt a message received in the form of a matrix $B$, they must calculate matrix $X$ such that:

$A \cdot X \cdot A^{-1} = B - 2I$

Given that $B = \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix}$, find matrix $X$, showing all intermediate steps.

Consider the matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ such that its determinant is $|A| = 1$. Prove that if the relation $A^2 - 5A + I = O$ holds (where $I$ is the identity matrix and $O$ is the zero matrix), then the inverse matrix $A^{-1}$ can be expressed as a linear combination of $A$ and $I$. Determine this expression and use it to find the inverse of $A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$ without using the adjugate matrix method.

In a market study on customer mobility between three phone operators ($P$, $Q$ and $R$), it has been determined that the transition matrix from one month to the next is $T$. If the current distribution of customers is the column matrix $C_{actual}$, and it is known that $T \cdot C_{anterior} = C_{actual}$, find the distribution from the previous month knowing that:

$T = \begin{pmatrix} 0,8 & 0,1 & 0,1 \\ 0,1 & 0,7 & 0,2 \\ 0 & 0,2 & 0,8 \end{pmatrix}$ and $C_{actual} = \begin{pmatrix} 4000 \\ 3500 \\ 2500 \end{pmatrix}$

Note: You must calculate the inverse of $T$ to go back in time.

Determine the rank of the following matrix $B$ depending on the values of the real parameters $a$ and $b$:

$B = \begin{pmatrix} 1 & a & 1 \\ 1 & 1 & b \\ 1 & 1 & 1 \end{pmatrix}$

For which values of $a$ and $b$ is matrix $B$ singular (has no inverse)? Illustrate the concept of linear dependence that occurs when the determinant is zero.

Consider the matrix $A = \begin{pmatrix} x & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & x \end{pmatrix}$.

1. a) Find the values of $x$ for which matrix $A + A^t$ does not have an inverse.
2. b) For $x = 1$, find a matrix $X$ that satisfies the equation $X \cdot A = A^2$.

Without expanding the determinant, use the properties to prove the following identity:

$\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} = (b-a)(c-a)(c-b)$

This is the famous Vandermonde determinant. Apply the result obtained to calculate the value of the determinant of the matrix:

$V = \begin{pmatrix} 1 & 1 & 1 \\ 10 & 20 & 30 \\ 100 & 400 & 900 \end{pmatrix}$

Consider a square matrix $A$ of order 3 whose determinant is $|A| = -0,5$.

1. a) Before calculating any definitive value, make a reasoned estimate: predict what sign they will have and whether their absolute value will be greater or less than 1 for the determinants $|A^{-1}|$ and $|\operatorname{Adj}(A)|$. Base your prediction on the theoretical relationships between determinants.
2. b) Using exclusively the properties of determinants (determinant of the product, of the inverse, of the adjugate, of the transpose, and of scalar multiplication), calculate the exact value of:
3. i) $|M_1|$, where $M_1 = 2 \cdot A^{-1}$

ii) $|M_2|$, where $M_2 = \operatorname{Adj}(A) \cdot (A^t)^{-1}$

1. c) Check whether the exact numerical results obtained in part b confirm the qualitative estimates of sign and order of magnitude you made in part a.

Consider the parametric matrix $A = \begin{pmatrix} 1 & 0 & k \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{pmatrix}$, where $k \in \mathbb{R}$.

1. a) Determine with reasoning for which values of parameter $k$ matrix $A$ is non-singular and, consequently, has an inverse matrix $A^{-1}$.
2. b) Without needing to calculate all entries of the inverse matrix, deduce the algebraic expression in terms of $k$ corresponding to the entry in the first row and third column of $A^{-1}$, that is, $(A^{-1})_{13}$, relying on the definition of the inverse using the transposed adjugate matrix. From this expression, find the value of $k$ for which $(A^{-1})_{13} = 1$.
3. c) For the value of $k$ found in part b, solve the matrix equation $A \cdot X - 2 \cdot I_3 = A^t$ with full justification, detailing each step in isolating the unknown $X$ and calculating its resulting matrix.