Problem 4. Prove the following by induction.
(a) For matrix $A = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}$, and all integers $n \ge 1$, $A^n = \begin{bmatrix} 1 & 3n \\ 0 & 1 \end{bmatrix}$.
(b) Given $n$ matrices where $n \ge 3$,
$A_1 \in \mathbb{R}^{d_1 \times d_2}$, $A_2 \in \mathbb{R}^{d_2 \times d_3}$, ..., $A_n \in \mathbb{R}^{d_n \times d_{n+1}}$,
the product $A_1 A_2 ... A_n$ is independent of how one places the parentheses.
