For two invertible $n \times n$
matrices $A$ and $B$, determine the formulas stated below is necessarily true:

$A + B$ is invertible, and$(A + B)^{-1} = A^{-1} + B^{-1}$

For two invertible $n \times n$
matrices $A$ and $B$, determine the formulas stated below is necessarily true.

$$(A + B)^2 = A^2 + 2AB + B^2$$

For two invertible $n \times n$
matrices $A$ and $B$, determine the formulas stated below is necessarily true.

$$(A − B)(A + B) = A^2 − B^2$$

A and B are two nxn matrices such that I-AB is invertibke. Show that I-BA is also invertible with inverse: I + B((I-AB)^-1)A

Consider two $n \times n$ matrices $A$ and $B$ such that the product $AB$ is invertible. Show that the matrices $A$ and B are both invertible. $Hint: AB(AB)^{-1} = I_n$ and $(AB)^{-1}AB = I_n$ . Use Theorem 2.4.8.

Let $L$ be the set of all strings, including the null string, that can be constructed by repeated application of the following rules: If $\alpha\in L$, then $a\alpha b\in L$ and $b\alpha a\in L$. If $\alpha\in L$ and $\beta\in L$, then $\alpha\beta\in L$.

Prove that if $\alpha$ has equal number of $a$'s and $b$'s, then $\alpha\in L$.

Consider a square matrix that differs from the identity matrix at just one entry, off the diagonal, for example,

$$\begin{bmatrix} 1 & 0 & 0\\ 0& 1 &0\\ -\frac{1}{2}& 0 &1 \end{bmatrix}$$

In general, is a matrix $M$ of this form invertible? If so, what is the $M^{−1}?$

Does the graph of $- 8 x ^ { 2 } - 12 = 3 - y$ open upward or downward?

Suppose that p(x) is the density function for heights of American men, in inches. What is the meaning of the statement p(68)=0.02?

Consider the matrices of the form

$$A=\begin{bmatrix} a&b\\ b&-a \end{bmatrix},$$

where a and b are arbitrary constants. For which values of a and b is$A^{−1} = A?$

Graph the function defined by $f(x)=\frac{-2}{x+1}$. Give the equation of its vertical asymptote.

Which lines are parallel? How do you know?

Use Figure the periodic table to list all the (a) inert gases, (b) alkalis, (c) halogens, and (d) alkaline earths.

A $20-\text{lb}$ weight is attached to the lower end of a coil spring suspended from a beam and comes to rest in its equilibrium position. The weight is then pulled down $1$ ft below its equilibrium position and released at $t=0$ with an initial velocity of $v_0\,\,\text{ft}/\text{sec}$, directed downward. Neglect the resistance of the medium. Determine the spring constant $k$ and the initial velocity $v_0$ if the amplitude of the resulting motion is $\sqrt{17}/4$ and the period is $\pi/4$.