## All Tasks

#### Identification of a matrix

In the table you will find information about group E of the UEFA Champions League 20/21, after the qualifying phase. Considering the data in the table, the matrix, $M$, representative of the results obtained by the 4 teams is given by the elements:

# Matrices

Modeling

13

#### Matrix multiplication

Let $A$ be a $2\times 3$ matrix with elements position $a_{ij}$ given by $a_{ij}=i^2- 2j$ and $B$ be a $3\times 4$ matrix with elements position $b_{ij}$ given by $b_{ij}=(-1)^j(2i-j)$.
Let $D=AB$.
Find the dimension of $D$ and compute the values $d_{ij}$, for each matrix entry's.

# Operation

Training

13

#### Invertible matrices

Consider the matices $A$ and $B$:
$A=\begin{bmatrix}
2 & -4 \\
-1 & 2 \\
\end{bmatrix}$
$B=\begin{bmatrix}
2 & -4 \\
-1 & -2 \\
\end{bmatrix}$.
Which of the following is true?

# Inverse

# Determinate

Training

13

#### Matrix rank

Consider the matrix $A=\begin{bmatrix}0&0&b&1\\1&3&0&b\\0&0&b&a\\2&0&6&0\end{bmatrix}$.
Using the concept of rank of a matrix, indicate the values of the parameters $a,b\in \mathbb{R}$, for which there is the inverse matrix of $A$.

# Matrices

Learning

13

#### Gauss elimination method

Applying the Gaussian elimination method, the equivalent row echelon form matrix of the matrix $A=\begin{bmatrix}1&0&1&-1\\-2&3&1&0\\0&1&2&-3\\2&2&2&4\end{bmatrix}$ is:

# Matrices

Training

13

#### Elementar operations

Let $A=\begin{bmatrix} 1&1&3\\1&-1&0\\0&2&4\end{bmatrix}$ be a real matrix $M_{3\times 3}$. Consider that the following elementary operations are performed on $A$, in the order presented:
(1) add to the 2nd column, the 3rd column multiplied by $3$
($c_2\leftarrow c_2 + 3c_3$);
(2) multiply the 3rd row by $\frac{1}{2}$ ($r_3\leftarrow \frac{1}{2}r_3$);
(3) exchange 2nd with 3rd rows ($r_2\leftrightarrow r_3$);
(4) add to the 3rd row, the additive inverse of the 1st row ($r_3\leftarrow r_3- r_1$);
(5) exchange 2nd with 3rd columns ($c_2\leftrightarrow c_3$).
The resulting matrix is:

# Matrices

Learning

13