Thread: Prove a binary operation is closed

1. Prove a binary operation is closed

If $\displaystyle a_1, a_2,b_1, b_2 \in \mathbb{R} , A_1,A_2 \in T$ where $\displaystyle A_1=\left(\begin{array}{cc}a_1&-b_1\\b_1&a_1\end{array}\right)$ and $\displaystyle A_2=\left(\begin{array}{cc}a_2&-b_2\\b_2&a_2\end{array}\right)$. Is $\displaystyle A_1+A_2$ necessarily a close operation on T?

2. Well, what do you think? When you add two matrices like this, is the result a matrix of the same form?

3. The proof in my book says, "Since $\displaystyle A_1,A_2 \in T$, then $\displaystyle A_1+A_2 \in T$. Therefor the operation is closed on $\displaystyle T$." I have very difficulties accepting it because somewhere else in the book says, "If $\displaystyle T \subset S$ and $\displaystyle a,b \in T, a*b$ is closed on $\displaystyle S$, but need not be closed on $\displaystyle T$.

4. Book example

Let $\displaystyle T=\{\left[\begin{array}{cc}a&-b\\b&a\end{array}\right]: a,b \in \mathbb{R}\}$. Is $\displaystyle T$ closed under matrix addition.

Proof:
Let $\displaystyle A_1, A_2 \in T$. Then $\displaystyle A_1=\left[\begin{array}{cc}a_1&-b_1\\b_1&a_1\end{array}\right]$ and $\displaystyle A_2=\left[\begin{array}{cc}a_2&-b_2\\b_2&a_2\end{array}\right]$ for some $\displaystyle a_1,b_1,a_2,b_2 \in \mathbb{R}$. Then
$\displaystyle A_1+A_2=\left[\begin{array}{cc}a_1+a_2&-(b_1+b_2)\\b_1+b_2&a_1+a_2\end{array}\right]$. Since $\displaystyle A_1+A_2\in T$, it follows that $\displaystyle T$ is closed under addition.

We know that $\displaystyle A_1+A_2 \in \mathbb{R}$, but how could it be possible that $\displaystyle A_1+A_2 \in T$ ?

Oh, yeah, never mind. I see now that if$\displaystyle A_1+A_2 \in \mathbb{R}$ then $\displaystyle A_1+A_2 \in T$ according to $\displaystyle T=\{\left[\begin{array}{cc}a&-b\\b&a\end{array}\right]: a,b \in \mathbb{R}\}$
Thanks again sir.

5. That looks right to me.