(a) If

$$A = \left[ \begin{array} { l l } { 1 } & { 1 } \\ { 0 } & { 1 } \end{array} \right]$$

, show that

$$U ^ { - 1 } A U$$

is not diagonal for any invertible complex matrix U. (b) If

$$A = \left[ \begin{array} { l l } { 0 } & { 1 } \\ { - 1 } & { 0 } \end{array} \right]$$

, show that

$$U ^ { - 1 } A U$$

is not upper triangular for any real invertible matrix U.

Show that if A is an invertible matrix, then

$$cond(A) \ge 1$$

with respect to any matrix norm.

Let A and B be $m\times n$ and $n\times m$ real matrices. (a)Prove that if $\lambda$ is a nonzero eigenvalue of the $m \times m$ matrix AB then it is also an eigenvalue of the $n\times n$ matrix BA. Show by example that this need not be true if $\lambda=0 .$ (b) Prove that $I_{m}-A B$ is invertible if and only if $I_{n}-B A$ is invertible.

Let A be an invertible matrix. If $\lambda$ is an eigenvalue of A show that $\lambda \neq 0$ and that $\lambda^{-1}$ is an eigenvalue of $A^{-1}$.

Let A and B be $n \times n$ matrices. a. Show that if A is invertible and AB = 0, then B = 0. b. If A is not invertible, show there is an $n \times n$ matrix B that is not the zero matrix and such that AB = 0.

Find the value of k so that the ordered pair satisfies the equation. 2x+y=k; (2, 1)

Use the test of your choice to determine whether the following series converge.

$$\sum _ { k = 1 } ^ { \infty } \left( \frac { 1 } { k } + 2 ^ { - k } \right)$$

Perform the indicated matrix operations or solve the matrix equation for X given that A, B, and C are defined as follows. If an operation is not defined, state the reason.

$$A = \left[ \begin{array} { r r } { 0 } & { 2 } \\ { - 1 } & { 3 } \\ { 1 } & { 0 } \end{array} \right] \quad B = \left[ \begin{array} { r r } { 4 } & { 1 } \\ { - 6 } & { - 2 } \end{array} \right] \quad C = \left[ \begin{array} { r r } { - 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right]$$

$$2 C - \frac { 1 } { 2 } B$$

What are the caveats to conducting t -tests on all of the individual

$$\beta$$

parameters in a multiple-regression model?

Compare how the respiratory, muscular, and nervous systems interrelate to accomplish the process of breathing. If there were a disruption in any one of these systems, how would it affect homeostasis in the body?

This exercise demonstrates that, in general, the results provided by Tchebysheff’s theorem cannot be improved upon. Let Y be a random variable such that

$$p ( - 1 ) = \frac { 1 } { 18 } , \quad p ( 0 ) = \frac { 16 } { 18 } , \quad p ( 1 ) = \frac { 1 } { 18 }$$

Show that E(Y ) = 0 and V(Y ) = 1/9.

Graph the function. state the domain and range.

$$p ( x ) = \frac { 3 } { x + 1 } - 2$$

Let X, Y, Z, W be i.i.d. positive r.v.s with CDF F and E(X) = 1. Write the most appropriate of $\leq, \geq, =,$ or ? in each blank (where “?” means that no relation holds in general).

$$\begin{array} { l } { \text { (a) } F ( 3 ) \_ 2 / 3 } \\ { \text { (b) } ( F ( 3 ) ) ^ { 3 } \_ P ( X + Y + Z \leq 9 ) } \\ { \text { (c) } E \left( \frac { X ^ { 2 } } { X ^ { 2 } + Y ^ { 2 } + Z ^ { 2 } + W ^ { 2 } } \right) \_ 1 / 4 \\ \text { (d) } E ( X Y Z W ) \_ E \left( X ^ { 4 } \right) } \\ { \text { (e) } \operatorname { Var } ( E ( Y | X ) ) \_ \operatorname { Var } ( Y ) } \\ { \text { (f) } \operatorname { Cov } ( X + Y , X - Y ) \_ 0} \end{array}$$

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined.

$$\cos ^ { - 1 } ( - 0.92761 )$$