Let $T: M_{n \times n} \to R$ be defined as $T(A) = det(A)$. Is $T$ linear? Explain!

Question

I'm having some trouble figuring out where to start with this problem. I know that for it to be a linear transformation it needs to satisfy two conditions $$t(u+v) = t(u) + t(v)$$ $$t(cu) = ct(u)$$

Thanks

Answer 1

Well for one thing you can compute$$\det\left(\begin{pmatrix}1&0\\0&0 \end{pmatrix}+\begin{pmatrix}0 &0\\0 &1\end{pmatrix} \right)$$

and then the sum of determinants.

Answer 2

Hint. Calculate the determinant of $I+I$ for the $2 \times 2$ identity matrix $I$.