Suppose that A and B are arbitrary $2 \times 2$ matrices. Is the quantity given always equal to $(A+B)^{2} ?$ $(B+A)^{2}$

Find an eigenvector associated with the given eigenvalue of A.

$$A=\left[\begin{array}{ll} {9} & {0} \\ {2} & {3} \end{array}\right] \quad \lambda=9$$

Suppose that A and B are arbitrary $2 \times 2$ matrices. Is the quantity given always equal to $(A+B)^{2} ?$ $A^{2}+2 A B+B^{2}$

Construct the matrix diagram for each of the following models. Yellow perch is a species of freshwater fish. Suppose a population of interest lives to a maximum age of 4 years. Only 5% of age 1 individuals survive to age 2, 20% of age 2 individuals survive to age 3, and 75% of age 3 individuals survive to age 4. Both age 3 and age 4 individuals reproduce, producing 100 and 150 age 1 offspring, respectively. Using $n_{i, t}$ for the number of individuals of age i at time t, we have

$$\begin{aligned} &n_{1, t+1}=100 n_{3, t}+150 n_{4, t}\\ &n_{2, t+1}=0.05 n_{1, t}\\ &n_{3, t+1}=0.2 n_{2, t}\\ &n_{4, t+1}=0.75 n_{3, t} \end{aligned}$$

If a $2 \times 2$ matrix has complex eigenvalues, what does this matrix do to vectors upon multiplication?