If a matrix A is $5 \times 3$ and the product AB is $5 \times 7$, what is the size of B?

Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot position in each column. Explain why the system has a unique solution.

Let

$$A=\left[ \begin{array}{rr}{2} & {5} \\ {-3} & {1}\end{array}\right]$$

and

$$B=\left[ \begin{array}{ll}{4} & {-5} \\ {3} & {k}\end{array}\right]$$

. What value(s) of k, if any, will make AB=BA?

If a matrix A is 5 × 3 and the product AB is 5 × 7, what isthe size of B?

Why is the question “Is the linear transformation T onto?” an existence question?

While being X-rayed, a person absorbs $3.2 \times 10^{-3} \mathrm{J}$ of energy. Determine the number of 40,000-eV X-ray photons absorbed during the exam.

Determine $h$ and $k$ such that the solution set of the system (i) is empty, (ii) contains a unique solution, and (iii) contains infinitely many solutions.

$$\begin{aligned}-2 x_1+h x_2 &=1 \\ 6 x_1+k x_2 &=-2 \end{aligned}$$

Let

$$A=\left[\begin{array}{rr} 2 & 3 \\ -1 & 1 \end{array}\right]$$

and

$$B=\left[\begin{array}{rr} 1 & 9 \\ -3 & k \end{array}\right]$$

. What value(s) of k, if any, will make AB = BA?

Use the random numbers to solve the following problem. Tankers arrive at an oil port with the distribution of interarrival times shown in Table 12. The port has two terminals, A and B. Terminal B is newer and therefore more efficient than terminal A. The time it takes to unload a tanker depends on the tanker’s size. A supertanker takes 4 days to unload at terminal A and 3 days at terminal B. A midsize tanker takes 3 days at terminal A and 2 days at terminal B. The small tankers take 2 days at terminal A and 1 day at terminal B. Arriving tankers form a single waiting line in the port area until a terminal becomes available for service. Service is given on an FCFS basis. The type of tankers and the frequency with which they visit this port is given by the distribution in Table 13. Develop a simulation model for this port. Compute such statistics as the average number of tankers in port, the average number of days in port for a tanker, and the percentage of idle time for each of the terminals. TABLE 12:

$$\begin{matrix} \text{Interarrival Times (days)} & \text{Probability}\\ \text{1} & \text{.20}\\ \text{2} & \text{.25}\\ \text{3} & \text{.35}\\ \text{4} & \text{.15}\\ \text{5} & \text{.05}\\ \end{matrix}$$

TABLE 13:

$$\begin{matrix} \text{Type of Tanker} & \text{Probability}\\ \text{Supertanker} & \text{.40}\\ \text{Midsize tanker} & \text{.35}\\ \text{Small tanker} & \text{.25}\\ \end{matrix}$$

View vectors in $\mathbb{R}^n$ as $n \times 1$ matrices. For u and v in $\mathbb{R}^n$, the matrix products $u^Tv$ is a $1 \times 1$ matrix, called the scalar product, or inner product, of u and v. It is usually written as a single real number without brackets. The matrix product $uv^T$ is an $n \times n$ matrix, called the outer product of u and v. If u and v are in $\mathbb{R}^n$, how are $u^Tv$ and $v^Tu$ related? How are $uv^T$ and $vu^T$ related?

Find or evaluate the integral. ∫ xe^(x²-1) dx

In the below Exercise, mark the statement True or False (T/F). Justify the answer.

(T/F) A mapping $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is one-to-one if each vector in $\mathbb{R}^n$ maps onto a unique vector in $\mathbb{R}^m$.

Let $S : \mathbb{R}^{p} \rightarrow \mathbb{R}^{n}$ and $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be linear transformations. Show that the mapping $\mathbf{x} \mapsto T(S(\mathbf{x}))$ is a linear transformation (from $\mathbb{R}^{p}$ to $\mathbb{R}^{m}$).

Use a rectangular coordinate system to plot

$$\mathbf{u}=\left[ \begin{array}{r}{5} \\ {2}\end{array}\right], \mathbf{v}=\left[ \begin{array}{r}{-2} \\ {4}\end{array}\right]$$

, and their images under the given transformation T. Describe geometrically what T does to each vector $\mathbf{x}$ in $\mathbb{R}^{2}$.