#### 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

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