Find the products $AB$ and $BA$ for the diagonal matrices.

$A=$

$$\begin{bmatrix} 3&0&0\\ 0&-5&0\\ 0&0&0 \end{bmatrix}$$

$B=$

$$\begin{bmatrix} -7&0 &0\\ 0&4 &0 \\ 0&0 &12 \end{bmatrix}$$

For arbitrary $2 \times 2$ matrices A, B, and C, verify the following properties. $A(B+C)=A B+A C$

Use the result in Exercise 160 above to determine the possible ranks for matrices of the form

$$\left[\begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & a_{16} \\ 0 & 0 & 0 & 0 & 0 & a_{26} \\ 0 & 0 & 0 & 0 & 0 & a_{36} \\ 0 & 0 & 0 & 0 & 0 & a_{46} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \end{array}\right]$$

Sketch a graph of a function with f'(x)0 for $x \neq 0$, f'(0) undefined, f''(x)>0 for x<0 and f''(x)<0 for x>0.

Determine whether the given matrix is in reduced row echelon form, row echelon form, or neither.

$$\left[\begin{array}{lllll}1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]$$

Sketch the graph of a function f for which f(0)=-1, f'(0)=0, f'(x)<0 if x<0, and f'(x)>0 if x>0.

Subdivide the transition matrix of each absorbing Markov chain

$$P=\left[\begin{array}{lllll} 1 & 0 & 0 & 0 & 0\\ 1/2 & 0 & 1/4 & 0 & 1/4\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 1/2 & 0 & 0 & 1/2 & 0\\ 0 & 1/2 & 1/2 & 0 & 0 \end{array}\right]$$

In exercise given below, can each matrix be an adjacency matrix?

$$\left[\begin{array}{lllll}0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]$$

Use the following matrices to compute the indicated expression if it is defined. A = [3 0, -1 2, 1 1], B = [4 -1, 0 2] C = [1 4 2, 3 1 5], D = [1 5 2, -1 0 1, 3 2 4], E = [6 1 3, -1 1 2, 4 1 3]. (a) (2DT - E)A (b) (4B)C + 2B (c) (-AC)T + 5DT (d) BAT - 2C)T (e) BT(CCT -ATA) (f) DTET -(ED)T

Find the matrix exponential of the matrix

$$\textbf{A}=\begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&1&0&0\\-1&0&0&0\end{bmatrix}$$

.

Prove that

$$\left[\begin{array}{lllll} 0 & 0 & 0 & a & 0 \\ 0 & 0 & b & 0 & c \\ 0 & d & 0 & e & 0 \\ f & 0 & g & 0 & 0 \\ 0 & h & 0 & 0 & 0 \end{array}\right]$$

is not invertible for any values of the entries.

Construct a directed graph whose adjacency matrix is equal to the given matrix.

$$\left[\begin{array}{llllll}{0} & {1} & {0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {0} & {0} & {0} \\ {0} & {0} & {0} & {1} & {0} & {1} \\ {0} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {1} & {0} & {0} \\ {1} & {1} & {0} & {0} & {0} & {0}\end{array}\right]$$

.

Find a row operation and the corresponding elementary matrix that will restore the given elementary matrix to the identity matrix. (a) [1 0, -3 1] (b) [1 0 0, 0 1 0, 0 0 3] (c) [0 0 0 1, 0 1 0 0, 0 0 1 0, 1 0 0 0] (d) [1 0 -1/7 0, 0 1 0 0, 0 0 1 0, 0 0 0 1]

$$\begin{aligned} &\text { Arterial Blood Gases (ABGs) }\\ &\begin{array}{ll} \mathrm{pH} & 7.28 \\ \mathrm{PaCO}_2 & 62 \mathrm{~mm} \mathrm{Hg} \\ \mathrm{HCO}_3 & 26 \mathrm{mmol} / \mathrm{L} \\ \mathrm{Pao}_2 & 48 \mathrm{~mm} \mathrm{Hg} \\ \mathrm{Spo}_2 & 53 \% \end{array} \end{aligned}$$

The ABG results from the sample drawn in the ED before intubation are sent to you. Interpret P.R.'s ABG results.