finding matrices to satisfy conditions

Question

Kind of a strange one, I believe the first one would be a matrix of 0 1 and 1 0 yet I can't find a way to compute these mathematically, but rather through trial and error.

Answer 1

Hint: Let $e_i$ be the vector whose $i$-th coordinate is $1$ and all other coordinates are $0$. For a compatible matrix $M$, what is $Me_i$?

Answer 2

We can for instance answer question b) by noting that $$ \pmatrix{y+z\\x+z\\x+y} = \pmatrix{0\cdot x + 1 \cdot y + 1 \cdot z\\ 1 \cdot x + 0 \cdot y + 1 \cdot z\\ 1 \cdot x + 1 \cdot y + 0 \cdot z} = \pmatrix{0&1&1\\1&0&1\\1&1&0}\pmatrix{x\\y\\z} $$

Answer 3

Hint:

For a), swapping row $i$ and row $j$ in a matrix amounts to left multiply it by the unit matrix, with rows $i$ and $j$ swapped.

For b), denoting $e_1,e_2,e_3$ the canonical basis, you have $$Ae_1=e_2+e_3,\quad Ae_2=e_3+e_1,\quad Ae_3=e_1+e_2.$$ Same method for c).