Exercises: Rank of a matrix

Calculate the determinant of the following matrix $A$ of order 2. This value represents the scale factor of a geometric transformation in the plane:

Find the determinant of the matrix $B$ using Sarrus' rule:

Given the matrix $M$, calculate the value of the real parameter $x$ so that its determinant is equal to $10$:

In the matrix $C$, identify the element $a_{23}$ and calculate its cofactor $A_{23}$:

Determine the rank of the following matrix $A$ by observing the relationship between its rows:

Given that the determinant of a square matrix $A$ of order 2 is $|A| = 5$, what is the value of the determinant of the matrix $2A$? Apply the properties of determinants to justify your answer.

Calculate the rank of the matrix $B$ by finding the highest-order minor that is nonzero:

1. a) State the highest possible order of a minor of $B$ and how many minors of that order can be formed.

1. b) Calculate one of those minors and check whether it is nonzero.

1. c) Write the rank of $B$ and justify your answer.

Quickly calculate the determinant of the following upper triangular matrix, giving reasons for your answer:

When calculating the rank of the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 0 & -1 \end{pmatrix}$, Marcos writes on his exam:

“We take the minor formed by the first two rows and the first two columns:

Since this minor of order 2 is zero, the matrix cannot have rank 2 or 3. Therefore, I conclude that $\operatorname{rg}(A) = 1$.”

1. a) Explain precisely Marcos's conceptual error: why does the fact that a specific minor of order 2 is zero not allow one to deduce that $\operatorname{rg}(A) < 2$?
2. b) Correct the exercise: locate a non-zero minor of order 2 in matrix $A$ and use the method of bordering minors to conclusively determine $\operatorname{rg}(A)$.

Given the matrix $B = \begin{pmatrix} 1 & -1 & 0 & 2 \\ 2 & -2 & 0 & 4 \\ 0 & 3 & 1 & -1 \end{pmatrix}$:

1. a) Estimate: before calculating any determinant, observe the rows of $B$ carefully. Justify with reasoning what the theoretical upper bound for $\operatorname{rg}(B)$ is based on its dimensions. Next, detect the evident proportional relationship between two of its rows and explain why the rank cannot be 3. From this visual inspection, make a reasoned estimate of whether the final rank will be 1 or 2.
2. b) Check: choose a non-zero minor of order 1 and apply the method of bordering minors. Find a non-zero minor of order 2 and, subsequently, calculate the determinants of all possible order 3 borderings to rigorously verify whether your initial estimate was correct.

Given a square matrix $A$ of order $3$ whose determinant is $|A| = 4$, use the properties of determinants to calculate with reasoning the value of the following determinants:

1. a) $|A^t \cdot A^{-1}|$

1. b) $|2 \cdot A \cdot A^t|$

1. c) The determinant of the matrix resulting from swapping the first two rows of $A$ and multiplying the third column by $-3$.

In the design of a graphics engine for a video game, a transformation matrix $M$ is required to be invertible in order to undo movements. Given the matrix:

$M = \begin{pmatrix} 1 & 0 & -1 \\ 0 & m & 3 \\ 4 & 1 & -m \end{pmatrix}$

1. a) Find the values of the real parameter $m$ for which the matrix $M$ does not possess an inverse.

1. b) For the value $m = 2$, calculate the value of the determinant of the inverse matrix, $|M^{-1}|$.

Determine the rank of the following matrix $A$ depending on the possible values of the real parameter $k$:

$A = \begin{pmatrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{pmatrix}$

Determine for which values of $k$ the rank is $1$, $2$ or $3$.

Knowing that the value of the determinant of the matrix $3 \times 3$ formed by the column vectors $\vec{c_1}$, $\vec{c_2}$ and $\vec{c_3}$ is equal to $5$, that is, $|\vec{c_1}, \vec{c_2}, \vec{c_3}| = 5$, calculate without expanding the value of:

$|\vec{c_1} + \vec{c_3}, \quad 2\vec{c_2}, \quad \vec{c_1} - \vec{c_3}|$

Justify each step mentioning the property of determinants that you have applied.

Calculate the value of the following determinant of order $4$ using Gaussian elimination to create zeros in a row or column before expanding by cofactors:

$|D| = \begin{vmatrix} 1 & 2 & 0 & 3 \\ -1 & 3 & 1 & 4 \\ 2 & 0 & -2 & 1 \\ 0 & 1 & 3 & 2 \end{vmatrix}$

Determine the rank of the matrix $B$ according to the values of the real parameter $a$:

$B = \begin{pmatrix} 1 & a & 1 & 2 \\ 2 & -1 & a & 1 \\ 1 & 10 & -3 & 5 \end{pmatrix}$

Remember that for a matrix of dimensions $3 \times 4$, the maximum possible rank is $3$.

Solve the following quadratic equation expressed in determinant form:

$\begin{vmatrix} x & 1 & 3 \\ 1 & x & 2 \\ 3 & 2 & 1 \end{vmatrix} = 0$

Find the exact solutions (they may contain roots).

Given the $3 \times 3$ matrices $A$ and $B$, you know that $A$ has rank $3$ and that the determinant of $B$ is $|B| = 0$.

1. a) What is the value of the determinant of the product $|A \cdot B|$? Justify your answer.

1. b) Is it possible for the rank of the product matrix $A \cdot B$ to be $3$? Explain why, based on the relationship between the determinant and the rank.

Consider the $3 \times 4$ matrix:

To determine the rank of $C$, two classmates propose different approaches:

• Strategy 1 (Gaussian elimination): apply elementary row operations until the matrix is in row echelon form.

• Strategy 2 (Bordering minors): discard zero columns and border a well-chosen non-zero minor of order 2.

1. a) Analyze the structure of matrix $C$ (pay special attention to the second column and the position of the zeros) and justify which of the two strategies requires fewer operations to argue the rank.
2. b) Following the strategy you justified as most convenient in the previous section, calculate $\operatorname{rg}(C)$ with reasoning, showing all steps and determinants or combinations used.

The State Meteorological Agency (AEMET) analyzes the accumulated quarterly precipitation (in $\text{l/m}^2$) at three stations in the Ebro basin: Logroño ($F_1$), Zaragoza ($F_2$), and Tortosa ($F_3$). The records corresponding to three consecutive quarters are collected in the matrix:

where the real parameter $k$ represents an isolated rainfall anomaly in Tortosa caused by a Mediterranean storm.

1. a) Model the linear dependence of the set by calculating the determinant of $P$ as a function of $k$. Determine for which exact value of $k$ the matrix $P$ is singular.
2. b) Interpret the two possible cases climatologically: calculate the rank of $P$ when $k = 0$ and when $k \ne 0$. Explain what each situation means in terms of meteorological information (does Tortosa provide independent data, or is its rainfall regime a predictable linear combination of Logroño and Zaragoza?).

Let $A$ be a square matrix of order $n=4$ such that its determinant is $|A| = \frac{1}{2}$. Using only the properties of determinants, calculate the value of the determinant of the matrix $B$, defined as:

where $A^t$ is the transpose of $A$ and $Adj(A)$ is its adjugate matrix. Justify each step of your calculation, indicating the property applied.

A robotics engineer is programming the movement of a mechanical arm in three-dimensional space. The sensors detect three force vectors that determine the stability of the system. The matrix representing these forces is:

You are asked to:

1. Determine the rank of the matrix $M$ in terms of the real parameter $k$.

2. Is there any value of $k$ for which the forces are linearly dependent but the rank of the matrix is maximum? Explain your answer based on the theory of minors.

Calculate, without expanding directly by Sarrus' rule, the value of the following determinant of order 4, applying elementary transformations to create zeros and simplify the calculation:

What condition must $a, b$ and $c$ satisfy for the rank of the associated matrix to be exactly 2?

Study the linear dependence of the rows of the matrix $A$ according to the values of the real parameters $a$ and $b$:

Determine for which values of $(a, b)$ the rank of $A$ is equal to 1. Describe geometrically what relationship the row vectors have when this occurs.

Given the square matrix of order 3:

1. Show that $|A|$ is independent of the value of the variable $x$.

2. If we define a new matrix $B = A - \lambda I$ (where $I$ is the identity), find the values of $\lambda$ for which the rank of $B$ is less than 3 when $x = \frac{\pi}{4}$.

In a telecommunications system, the signal is lost if the determinant of the transmission matrix vanishes. The transmission matrix is given by:

1. a) Knowing that $|T|$ is a Vandermonde determinant, express it as a product of linear factors.

1. b) If we set $a=2$ and $b=5$, discuss the rank of $T$ according to the values the parameter $m$ can take. For which values of $m$ is the rank minimal, and what is that minimum rank?

Consider the square matrix $A$ of order 3 whose columns are $C_1, C_2, C_3$. It is known that $|A| = 5$. Calculate the value of the determinant of the matrix $M$ whose columns are:

If we replace the third column of $M$ with the zero vector, what would be the rank of the new matrix? Justify your answer without performing detailed numerical calculations.

Given the matrix:

1. Find the values of $\alpha$ for which the matrix does not have an inverse.

2. For the values found in the previous section, calculate the rank of the matrix $A^2$.

3. Is it possible to find a value of $\alpha$ such that $rg(A) \neq rg(A^2)$? Justify your answer based on the properties of rank and determinants.

Design with reasoning a matrix $M \in \mathcal{M}_{3 \times 4}(\mathbb{R})$ that simultaneously satisfies the following three conditions:

1. The four entries of the first row are strictly positive and distinct integers: $a_{11} < a_{12} < a_{13} < a_{14}$.

2. The minor of order 2 formed by rows 1 and 2 and columns 1 and 2 is zero, that is, $\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = 0$, but there exists at least one non-zero minor of order 2 that contains row 1 and column 1.

3. $\operatorname{rg}(M) = 2$.

To justify your deductive construction:

• Explain how you determine rows 2 and 3 to ensure the required linear dependence and independence.

• Formally prove that $\operatorname{rg}(M) = 2$ using the method of bordering minors, stating the non-zero minor of order 2 from which you start and verifying that all its possible order 3 borderings have determinant equal to zero.

In a 2nd year of Bachillerato exam, students are asked to discuss the rank of the following matrix as a function of the real parameter $a$:

David presents the following solution:

“I calculate the determinant of the matrix and factor it:

I set it equal to zero and obtain the roots $a = 1$ and $a = -2$.

• Case 1: If $a \ne 1$ and $a \ne -2$, $|A_a| \ne 0$, so the matrix is non-singular and $\operatorname{rg}(A_a) = 3$.

• Case 2: If $a = 1$ or $a = -2$, since $|A_a| = 0$, the matrix cannot have rank 3; therefore, in both cases $\operatorname{rg}(A_a) = 2$.”

1. a) Locate David's deductive reasoning error in Case 2. Why does the condition $|A| = 0$ in a square matrix of order $n$ only allow one to state that $\operatorname{rg}(A) \le n - 1$ and not guarantee that $\operatorname{rg}(A) = n - 1$?
2. b) Correct the study for $a = 1$ and for $a = -2$. For each of these two values, determine the exact rank of the matrix by analyzing the existence or complete vanishing of minors of order 2.
3. c) Based on your conclusions, justify for which values of $a$ the inverse matrix $A_a^{-1}$ exists, and relate this result to the definition of an invertible (regular) matrix.