## All Tasks

#### Invertible matrix - using rank

Let $A$ be an $n$ by $n$ matrix. Knowing that $A$ has an inverse iff $rank(A) = n$, determine if the matrix $C=\begin{bmatrix}
1&0&-3\\0&-\frac{1}{3}&\ \frac{1}{3}\\1&-\frac{2}{3}&-\frac{7}{3}
\end{bmatrix}$ is invertible.

# Matrices

Learning

13

#### Rank of a matrix

Consider the matrix $M=\begin{bmatrix} 2&a&0\\-1&0&-2\\b&-1&2\end{bmatrix}$. Find a relation for the real constants $a$ and $b$ so that rank of $M$ is equal to 2.

# Matrices

Training

13

#### What is the total cost?

João ate a fruit salad that contained x portions of pineapple, y portions of mango and z portions of pear (1 portion is 100g fruit). Matrix A, represents the amounts of energy (calories), iron (mg), and calcium (mg) and Matrix C indicates the prices (euros), of each portion of the 3 fruits. Matrix B shows what João ingested in total. What is the cost of this fruit salad?
$A=\begin{bmatrix}
52&64&39\\
0.5&0.8&0.9\\
18&21&22\\
\end{bmatrix}$ $\begin{array}{l}
calories \\
iron\\
calcium\\
\end{array}$,
$B=\begin{bmatrix}
246 \\3.6\\101\\
\end{bmatrix}$ $\begin{array}{l}
calories \\
iron\\
calcium\\
\end{array}$, $ C=\begin{bmatrix}
0.10\\
0.30\\
0.25\\
\end{bmatrix}$ $\begin{array}{l}
pineapple \\
mango\\
pear\\
\end{array}$

# Matrices

# System of linear equations

Modeling

13

#### Matrices multiplication

Multiply the two matrices and enter the values $a$ to $f$ in the corresponding checkbox.
$\begin{bmatrix}
4 & 2 & 1 \\
0 & -2 & 4 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
1 & 2 & 2 \\
1 & 3 & 3 \\
1 & -1 & 1 \\
\end{bmatrix}
=
\begin{bmatrix}
a & b & c \\
d & e & f \\
\end{bmatrix}$

# Operation

Training

13

#### Matrix add

To illustrate the letter “A” in figures 1 and 2, 20 pixels in a 5×4 grid (for each figure) were used. The color of each pixel can be represented by a specific number, defined by the color scale.
If 1 and 2 can be represented by the matrices $M$ and $N$, determine the matrix $P$ which represents the contrast of the letter “A” when dark gray is changed to light gray and white is changed to black.

# Operation

Modeling

13

#### Total cost (I_M)

A factory, in the production of pieces, has a fixed cost of $8$ euros plus a variable cost of $0.50$ euros per unit produced. If $x$ is the number of pieces produced, what is the cost of $100$ pieces?

# Linear functions

Modeling

9