All Tasks - Asymptote

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 $A$.

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:

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:

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.

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

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}$ $\begin{array}{l} pineapple \\ mango\\ pear\\ \end{array}$