Let $M_{n,n}(\mathbb{R})$ denote the vector space of $n \times n$ matrices with real entries. Let $f : M_{2,2}(\mathbb{R}) \to \mathbb{R}$ be the function defined by $f(A) = \mathrm{det}(A)$ for any $A \in M_{2,2}(\mathbb{R})$. Is $f$ a linear transformation?

Let $A = \left[\begin{array}{cc} a_{11} &a_{12}\cr a_{21} &a_{22}\cr \end{array}\right]$ and $B = \left[\begin{array}{cc} b_{11} &b_{12}\cr b_{21} &b_{22}\cr \end{array}\right]$ be any two matrices in $M_{2,2}(\mathbb{R})$ and let $c \in \mathbb{R}$.

1. $f(A+B) =$ . (Enter $a_{11}$ as $\verb+a11+$, etc.)
   $f(A) + f(B) =$ $+$ .
   Does $f(A+B) = f(A) + f(B)$ for all $A, B \in M_{2,2}(\mathbb{R})$? choose Yes, they are equal No, they are not equal
   
   
2. $f(c A) =$ .
   $c(f(A)) =$ $\Big($ $\Big)$.
   Does $f(c A) = c(f(A))$ for all $c \in \mathbb{R}$ and all $A \in M_{2,2}(\mathbb{R})$? choose Yes, they are equal No, they are not equal
   
   
3. Is $f$ a linear transformation? choose f is a linear transformation f is not a linear transformation