How is this M(2,2)->R closed under addition? 
so I understand that It is closed under scalar multiplication, but how is it closed under addition? what if I add another matrix to A? and create new a,b,c,d values?
 A: Help me fill this in:
\begin{align*}
T \begin{pmatrix}a_1 & b_1 \\ c_1 & d_1 \end{pmatrix} &= \underline{\hspace{50pt}} \in \mathbb{R} \\
T \begin{pmatrix}a_2 & b_2 \\ c_2 & d_2 \end{pmatrix} &= \underline{\hspace{50pt}} \in \mathbb{R} \\
T \begin{pmatrix}a_1 & b_1 \\ c_1 & d_1 \end{pmatrix} +T \begin{pmatrix}a_2 & b_2 \\ c_2 & d_2 \end{pmatrix} &= \underline{\hspace{50pt}} \in \mathbb{R} \\
\begin{pmatrix}a_1 & b_1 \\ c_1 & d_1 \end{pmatrix} + \begin{pmatrix}a_2 & b_2 \\ c_2 & d_2 \end{pmatrix} &= \underline{\hspace{50pt}} \in M_{2, 2} \\
T\left(\begin{pmatrix}a_1 & b_1 \\ c_1 & d_1 \end{pmatrix} + \begin{pmatrix}a_2 & b_2 \\ c_2 & d_2 \end{pmatrix}\right) &= \underline{\hspace{50pt}} \in \mathbb{R}.
\end{align*}
Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.
A: TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.
