Rank of a matrix
A market research company records the answers to several survey questions in a matrix, one row per question. It may happen that one of the questions is, deep down, a combination of the others: it provides no new information. Knowing how many rows (or columns) of a matrix are truly "independent" is exactly what the rank measures.
What the rank of a matrix measures
If one row is a multiple of another, or is the sum of two others, it does not add a new direction: it is "redundant" as far as rank is concerned, even though it still takes up a row of the matrix.
In a digital marketing campaign, a company analyses the impact of two social media ads. The results are organised in a matrix…
In a digital marketing campaign, a company analyses the impact of two social media ads. The results are organised in a matrix $A = \begin{pmatrix} 120 & 45 \\ 80 & 30 \end{pmatrix}$, where the first column represents the "clicks" and the second the "purchases" made. The first row corresponds to ad 1 and the second to ad 2. The figure represents each ad as the vector (clicks, purchases) from the origin.
- a) Calculate the determinant of the matrix $A$.
- b) Explain whether there is any proportionality relationship between the data of the two ads and how it relates to the value of the determinant and to the figure.
Rank using minors
A student is designing an algorithm for a video game and needs a transformation matrix to be "singular" (that is, its determinant is equal to zero). If the matrix is…
A student is designing an algorithm for a video game and needs a transformation matrix to be "singular" (that is, its determinant is equal to zero). If the matrix is $M = \begin{pmatrix} x & 4 \\ 9 & x \end{pmatrix}$, find the possible real values of $x$ so that condition $|M| = 0$ is satisfied.
Calculate the value of the following 3rd order determinant using Sarrus' rule: $$\begin{vmatrix} 1 & 0 & 3 \\ 2 & 1 & 4 \\ 1 & 0 & 2 \end{vmatrix}$$ State step-by-step…
Calculate the value of the following 3rd order determinant using Sarrus' rule:
$$\begin{vmatrix} 1 & 0 & 3 \\ 2 & 1 & 4 \\ 1 & 0 & 2 \end{vmatrix}$$State step-by-step the products of the main diagonals and the secondary diagonals.
The method of bordered minors
Consider matrices $A$ and $B$ such that: $$A = \begin{pmatrix} 2 & 3 \\ 4 & 6 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 2 & -1 \end{pmatrix}$$ a) Calculate $|A|$…
Consider matrices $A$ and $B$ such that:
$$A = \begin{pmatrix} 2 & 3 \\ 4 & 6 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 2 & -1 \end{pmatrix}$$- a) Calculate $|A|$ and $|B|$.
- b) Calculate $|A \cdot B|$ without performing the matrix multiplication, justifying which property you use.
- c) Is the matrix $A$ invertible? What about the matrix $B$? Justify your answer based on the value of their determinants.
A cybersecurity company uses matrices to validate access keys. For a key to be secure, the rank of its associated matrix must be maximum. Consider the key matrix $K$:…
A cybersecurity company uses matrices to validate access keys. For a key to be secure, the rank of its associated matrix must be maximum. Consider the key matrix $K$:
$$K = \begin{pmatrix} 1 & -1 & k \\ k & 1 & -1 \\ -1 & k & 1 \end{pmatrix}$$-
Calculate the values of $k$ for which the key is not secure (that is, the rank of $K$ is less than 3).
-
If $k = -1$, study the linear dependence of the rows of the matrix. What is the rank in this case?
-
For the values of $k$ such that $|K| = 0$, state whether there exists any minor of order 2 that is non-zero.
Rank according to a parameter
When the matrix depends on a letter $k$, the rank can change depending on the value it takes. The strategy is always the same: calculate the largest possible determinant as a function of $k$ (by Laplace expansion or by Gaussian elimination), factorise it and study what happens at its roots.
Determine the rank of the following matrix $M$ depending on the value of the real parameter $k$: $$M = \begin{pmatrix} 1 & k & 1 \\ 2 & 1 & k \\ 0 & 1 & 1…
Determine the rank of the following matrix $M$ depending on the value of the real parameter $k$:
$$M = \begin{pmatrix} 1 & k & 1 \\ 2 & 1 & k \\ 0 & 1 & 1 \end{pmatrix}$$State for which values of $k$ the rank is maximum and what happens in the critical cases.
Given the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix}$, calculate $|A|$ and, using the properties of determinants, find $|A^T|$,…
Given the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix}$, calculate $|A|$ and, using the properties of determinants, find $|A^T|$, $|A^{-1}|$, and $|A^2|$.
Rank, determinant and the product of matrices
Given that the determinant of a square matrix $A$ of order 3 is $|A| = -2$, use the properties of determinants to calculate, giving reasons, the value of: a) $|A^t|$…
Given that the determinant of a square matrix $A$ of order 3 is $|A| = -2$, use the properties of determinants to calculate, giving reasons, the value of:
- a) $|A^t|$ (determinant of the transpose).
- b) $|A^{-1}|$ (determinant of the inverse).
- c) $|3A|$.
- d) The determinant of the matrix resulting from swapping the first two columns of $A$ and multiplying the third row by $-5$.
Back to the beginning: in the survey data shown in the figure, the second question is proportional to the first ($F_2 = \frac{2}{3} \cdot F_1$), so there are only $2$ questions with independent answers and the rank of the matrix is $2$.