For ordinary arithmetic $2 \times 3=3 \times 2$ and in algebra ab = ba. For matrices, does AB always equal BA?

The notation $A^k$ means the matrix A multiplied with itself k times. For the $n \times n$ identity matrix I, show that $I^{2}=I.$

A certain oxide of titanium is 28.31% oxygen by mass and contains a mixture of $\mathrm { Ti } ^ { 2 + }$ and $\mathrm { Ti } ^ { 3 + }$ ions. Determine the formula of the compound and the relative numbers of $\mathrm { Ti } ^ { 2 + }$ and $\mathrm { Ti } ^ { 3 + }$ ions.

Find n for the given value of $a _ { n }$ in each arithmetic sequence. $a _ { n } = 95$; -1, 3, 7, ...

Each of the lettered choices below refers to the following numbered statements. Select the best lettered choice. A choice may be used once, more than once, or not at all. (A) Fitness (B) Single-gene trait (C) Polygenic trait (D) Hardy-Weinberg principle (E) Gene pool. 7. Survival and reproduction of individuals best suited to their environment.

A matrix $A$ is given. Find $A^k$ for an arbitrary positive integer $k$.

$$\begin{bmatrix}5&&-6\\3&&-4\end{bmatrix}$$

What is the process for a bill to become law?

A matrix $A$ is given. Find $A^k$ for an arbitrary positive integer $k$.

$$\begin{bmatrix}11&&8\\-12&&-9\end{bmatrix}$$

State whether the entropy of the system increases or decreases in each of the following processes: Pure gases are mixed to prepare an anesthetic.

Differentiate the given function and simplify your answer. $h(s)=(1+\sqrt{3} s)^{5}$

For each of the following statements, prove the statement if it is true or give a counterexample if it is false. a. The midpoint of the common chord of two circles that intersect in two points lies on the line through the centers of the circles. b. If inscribed angle $\angle A B C$ in a circle with center O measures $120^{\circ}$, then central angle $\angle A O C$ measures $120^{\circ}$. c. The angle bisector of any angle inscribed in a circle contains the center of the circle. d. If the line through the midpoints of two chords of a circle contains the center of the circle, then the chords are parallel.

Show that the covariance between $\hat { \alpha }$ and $\hat { \beta }$ is zero.

Use the multiplication table. a. Which numbers from 0 to 150 appear most frequently as entries in the multiplication table?
b. Which numbers from 0 to 150 appear least frequently as entries in the multiplication table?

Find each product.

$$7(-6)$$