Equivalence relation for set given as matrix

Question

I need a hint. My task is to proof that

• ((a11, a12), (b11, b12))

• ((a21, a22), (b21, b22))

  ∈ R ⇔ a11 + a12 + a21 + a22 = b11 + b12 + b21 + b22

R is equivalnce relation. My problem is that I have no idea how to deal with matrix.

I tried to work like in this post and just make something like A={{a11,a12},{a21,a22}} instead of A={0,1}, but I got stuck.

I would be grateful for any help.

Answer 1

Hint:

If $f:X\rightarrow Y$ is a function then the relation $R$ on $X$ defined by $xRy\iff f(x)=f(y)$ is an equivalence relation. So find a proper function here on $2\times 2$-matrices.

(Why? Simply because $f(x)=f(x)$ i.e. reflexitivity, $f(x)=f(y)\Rightarrow f(y)=f(x)$ i.e. symmetry, and $f(x)=f(y)\wedge f(y)=f(z)\Rightarrow f(x)=f(z)$ i.e. transitivity)