Problem $1.$ You receive the following message on Telegram from the user 'thelrueelonmusk'. thetrueelonmusk:

Dear user, I am happy to share that I finally managed to discover a loophole in the world's financial system: a formula that allows to generate infinite money! I am eager to share this with you as a token of gratitude for your support. To make using the formula easier, I created the website 'absolutelynotsuspicious.tv$\text{'}$ that will allow you to use my infinite money glitch. You just need to connect you crypto account to the website, add some funds and let the formula do the work for you!

You decide to follow the link $($on a virtual machine, of course$),$ and, after inspecting the source code of the webpage, you uncover the formula so carefully hidden by thetrueelonmusk. It is an exchange of cryplocurrency with the following rale $($USDT$,$ ETH and TON are cryplocurrency names, and the problem does not require to know their monetary value$)$:

The formula seems to work in the following way: it takes all the USDT, ETH and TON that you have on your account, and converts them all at once using the exchange. The website code then automatically applies the formula three total times $($each time, to the new balance after the previous application$)$ to maximise your gains. You seem to discover the scam: having any TON would just result in a net loss in the exchange. You therefore decide to launch the website code for an account with your ETH and USDT savings: $1$ ETH and $10$ USDT.

$(1)$ Did you profit from the operation?

$(2)$ What is the linear transformation underlying applying the formula? Provide its matrix $\backslash ($ A $\backslash ).$

$(3)$ Is A injective? Is A surjective?

$(4)$ What is the kernel of $\backslash ($ A $\backslash )?$ What is the image of $\backslash ($ A $\backslash )?$

$(5)$ What is the linear transformation underlying applying the formula twice? three times? What do you observe? In context of the problem, and assuming that the code ignores that balance cannot be negative while applying the formula repeatedly, which initial $($non$-$negative in each currency$)$ account balances could have led to profit?