Exercise. 7. Use the given partitions to compute the matrix products. Check your work by computing the
same products without using a partition. Show all your work.
$\begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ \hline 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 & 1 \\ 3 & 4 & 5 & 2 \\ \hline 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 4 \end{bmatrix}$
Exercise. 8. For the matrix A, compute $A^2 = AA$, $A^3 = AAA$, and $A^4$. Describe the pattern that emerges,
and use this pattern to find $A^{1001}$. Interpret your answers geometrically, in terms of rotations, reflections,
shears, and orthogonal projections.
$\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$
Exercise. 9. Find all matrices X that satisfy the given matrix equation.
$X \begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
