Compare and contrast the structures and chemistries of the hydrides of the group 14 elements, and give pertinent examples to illustrate structural and chemical differences between $\mathrm{BH}_3$ and $\mathrm{CH}_4$, and between $\mathrm{AlH}_3$ and $\mathrm{SiH}_4$.

Use Cramer's Rule to solve each system of equations.

• a - 2b + c = 7
• 6a + 2b - 2c = 4
• 4a + 6b + 4c = 14

$$\left. \begin{array} { l } { \text { Suppose that } a \text { and } b \text { are group elements that commute and have } } \\ { \text { orders } m \text { and } n . \text { If } \langle a \rangle \cap \langle b \rangle = \{ e \} , \text { prove that the group contains an } } \\ { \text { element whose order is the least common multiple of } m \text { and } n . } \\ { \text { Show that this need not be true if } a \text { and } b \text { do not commute. } } \end{array} \right.$$

Tell which solutions, if any, are extraneous for each equation.

$$2 a=\sqrt{4a+3} ; a=\frac{3}{2}, a=-\frac{1}{2}$$

How many elements are in A₁ ∪ A₂ if there are 12 elements in A₁, 18 elements in A₂, and a) A₁ ∩ A₂ = ∅? b) |A₁ ∩ A₂| = 1? c) |A₁ ∩ A₂| = 6? d) A₁ ⊆ A₂?

Let P be a partition of a group G with the property that for any pair of elements A, B of the partition, the product set AB is contained entirely within another element C of the partition. Let N be the element of P that contains 1. Prove that N is a normal subgroup of G and that P is the set of cosets.

Let G be a group and let S be the set of all elements of the form $a b a^{-1} b^{-1}$ with a, $b \in G$. The subgroup $G^{\prime}$ generated by the set S is called the commutator subgroup of G. Prove (a) $G^{\prime}$ is normal in G. (b) $G / G^{\prime}$ is abelian.

Here you need to show that if a square matrix $A$ satisfies the given condition:

$$A^3+4A^2-2A=7I=0$$

then so does $A^T.$

a rectangle with corners labeled 1 through 4 is placed in a reference rectangle We can form a group in which each element is a rotation of the inner rectangle that allows us to reposition the inner rectangle within the reference rectangle, similar to the symmetry groups discussed in earlier Section. We again use the “followed by” operation, $\triangle$.

Explain why this group has only four elements.

A group G of order $n \geq 2$ contains a subgroup H. In a left coset decomposition of H in G, the number of distinct left cosets is the same as the order of H. If some left coset contains p elements, where p is a prime, then what is n?

solve for X in the equation, where A = [-4 0 , 1 -5 , -3 2] and B=[1 2 , -2 1 , 4 4]. 6X = 4A + 3B

Identify the number of terms, and then the terms themselves for each algebraic expression. a. 4 - 3x b. 4a - 9 + 3a c. 7b - 9x + 3a - 12

Use Cramer's Rule to solve each system of equations.

• 3a - 5b + 2c = -5
• 4a + b + 3c = 9
• 2a - c = 1

Determine the values of the literal numbers.

$$\left[\begin{array}{c} 4a \\ a-b \end{array}\right]=\begin{bmatrix} 8 \\ 5 \end{bmatrix}$$