Rank of a matrix

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

$$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 0 & 1 & 1 \end{pmatrix}$$

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

$$B = \begin{pmatrix} 1 & 0 & 5 \\ -2 & 1 & 3 \end{pmatrix}$$

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.

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

$$C = \begin{pmatrix} 2 & 1 & 5 \\ 0 & -1 & 3 \\ 4 & 2 & 1 \end{pmatrix}$$

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

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

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 $3$.

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:

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

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.

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 $3$.

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

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

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.

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, $|M^{-1}|$.

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.