Math Problems Vault

Problem 3.5.1

For $\mathbf{A} = \begin{bmatrix} 1 & -2 & 3 \\ 0 & -5 & 4 \\ 4 & -3 & 8 \end{bmatrix}$ , $\mathbf{B} = \begin{bmatrix} 1 & 2 \\ 0 & 4 \\ 3 & 7 \end{bmatrix}$ , and $\mathbf{C} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ , compute the following products when possible.

?

(a)

$\mathbf{A}\mathbf{B}$ .

(b)

$\mathbf{B}\mathbf{A}$ .

(c)

$\mathbf{C}\mathbf{B}$ .

(d)

$\mathbf{C}^{T}\mathbf{B}$ .

(e)

$\mathbf{A}^{2}$ .

(f)

$\mathbf{B}^{2}$ .

(g)

$\mathbf{C}^{T}\mathbf{C}$ .

(h)

$\mathbf{C}\mathbf{C}^{T}$ .

(i)

$\mathbf{B}\mathbf{B}^{T}$ .

(j)

$\mathbf{B}^{T}\mathbf{B}$ .

(k)

$\mathbf{C}^{T}\mathbf{A}\mathbf{C}$ .

Problem 3.5.2

(a)(b)(c)

Consider the following system of equations:

\begin{aligned} 2x_{1} + x_{2} + x_{3} & = 3, \\ 4x_{1} + 2x_{3} & = 10, \\ 2x_{1} + 2x_{2} & = -2. \end{aligned}

?

(a)

Write the system as a matrix equation of the form $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ .

(b)

Write the solution of the system as a column $\overrightarrow {s}$ and verify by matrix multiplication that $\overrightarrow {s}$ satisfies the equation $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ .

(c)

Write $\overrightarrow {b}$ as a linear combination of the columns in $\mathbf{A}$ .

Problem 3.5.3

(a)(b)

Let $\mathbf{E} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 3 & 0 & 1 \end{bmatrix}$ and let $\mathbf{A}$ be an arbitrary $3 \times 3$ matrix.

?

(a)

Describe the rows of $\mathbf{E}\mathbf{A}$ in terms of the rows of $\mathbf{A}$ .

(b)

Describe the columns of $\mathbf{A}\mathbf{E}$ in terms of the columns of $\mathbf{A}$ .

Problem 3.5.4

(a)(b)(c)

Let $\overrightarrow {e}_{j}$ denote the $j^{\mathrm{th}}$ unit column that contains a $1$ in the $j^{\mathrm{th}}$ position and zeros everywhere else. For a general matrix $\mathbf{A}_{n \times n}$ , describe the following products.

?

(a)

$\mathbf{A}\overrightarrow {e}_{j}$ .

(b)

$\overrightarrow {e}_{i}^{T}\mathbf{A}$ .

(c)

$\overrightarrow {e}_{i}^{T}\mathbf{A}\overrightarrow {e}_{j}$ .

Problem 3.5.5

Suppose that $\mathbf{A}$ and $\mathbf{B}$ are $m \times n$ matrices. If $\mathbf{A}\overrightarrow {x} = \mathbf{B}\overrightarrow {x}$ holds for all $n \times 1$ columns $\overrightarrow {x}$ , prove that $\mathbf{A} = \mathbf{B}$ .

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Problem 3.5.6

For $\mathbf{A} = \begin{bmatrix} 1/2 & \alpha \\ 0 & 1/2 \end{bmatrix}$ , determine $\lim_{n \to \infty } \mathbf{A}^{n}$ .

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Problem 3.5.7

If $\mathbf{C}_{m \times 1}$ and $\mathbf{R}_{1 \times n}$ are matrices consisting of a single column and a single row, respectively, then the matrix product $\mathbf{P}_{m \times n} = \mathbf{C}\mathbf{R}$ is sometimes called the outer product of $\mathbf{C}$ with $\mathbf{R}$ . For conformable matrices $\mathbf{A}$ and $\mathbf{B}$ , explain how to write the product $\mathbf{A}\mathbf{B}$ as a sum of outer products involving the columns of $\mathbf{A}$ and the rows of $\mathbf{B}$ .

?

Problem 3.5.8

(a)(b)(c)

A square matrix $\mathbf{U} = [u_{ij}]$ is said to be upper triangular whenever $u_{ij} = 0$ for $i > j$ -- i.e., all entries below the main diagonal are $0$ .

?

(a)

If $\mathbf{A}$ and $\mathbf{B}$ are two $n \times n$ upper-triangular matrices, explain why the product $\mathbf{A}\mathbf{B}$ must also be upper triangular.

(b)

If $\mathbf{A}_{n \times n}$ and $\mathbf{B}_{n \times n}$ are upper triangular, what are the diagonal entries of $\mathbf{A}\mathbf{B}$ ?

(c)

$\mathbf{L}$ is lower triangular when $\ell_{ij} = 0$ for $i < j$ . Is it true that the product of two $n \times n$ lower-triangular matrices is again lower triangular?

Problem 3.5.9

If $\mathbf{A} = [a_{ij}(t)]$ is a matrix whose entries are functions of a variable $t$ , the derivative of $\mathbf{A}$ with respect to $t$ is defined to be the matrix of derivatives. That is,

\frac{d\mathbf{A}}{dt} = \left[\frac{da_{ij}}{dt}\right].

Derive the product rule for differentiation

\frac{d(\mathbf{A}\mathbf{B})}{dt} = \frac{d\mathbf{A}}{dt}\mathbf{B} + \mathbf{A}\frac{d\mathbf{B}}{dt}.

?

Problem 3.5.10

(a)(b)

Recall Example 3.5.2's construction: for a network of $n$ nodes (e.g. cities served by an airline), the connectivity matrix $\mathbf{C}_{n \times n} = [c_{ij}]$ is defined by $c_{ij} = 1$ if there is a direct path (flight) from node $i$ to node $j$ , and $c_{ij} = 0$ otherwise. Let $\mathbf{C}_{n \times n}$ be the connectivity matrix associated with such a network of $n$ nodes, and let $\overrightarrow {e}$ be the $n \times 1$ column of all $1$ 's. In terms of the network, describe the entries in each of the following products.

?

(a)

Interpret the product $\mathbf{C}\overrightarrow {e}$ .

(b)

Interpret the product $\overrightarrow {e}^{T}\mathbf{C}$ .

Problem 3.5.11

Consider three tanks each containing $V$ gallons of brine. The tanks are connected in a line, and all spigots are opened at once. As fresh water at the rate of $r$ gal/sec is pumped into the top of the first tank, $r$ gal/sec leaves from the bottom and flows into the next tank, and so on down the line -- there are $r$ gal/sec entering at the top and leaving through the bottom of each tank.

Figure 3.5.2

Let $x_{i}(t)$ denote the number of pounds of salt in tank $i$ at time $t$ , and let

\overrightarrow {x} = \begin{pmatrix} x_{1}(t) \\ x_{2}(t) \\ x_{3}(t) \end{pmatrix} \qquad \text{and} \qquad \frac{d\overrightarrow {x}}{dt} = \begin{pmatrix} dx_{1}/dt \\ dx_{2}/dt \\ dx_{3}/dt \end{pmatrix}.

Assuming that complete mixing occurs in each tank on a continuous basis, show that

\frac{d\overrightarrow {x}}{dt} = \mathbf{A}\overrightarrow {x}, \qquad \text{where} \qquad \mathbf{A} = \frac{r}{V}\begin{bmatrix} -1 & 0 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{bmatrix}.

?

Remark.

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Use the fact that $\dfrac {dx_{i}}{dt} = \text{rate of change} = \dfrac {\text{lbs}}{\text{sec}} \text{ coming in} - \dfrac {\text{lbs}}{\text{sec}} \text{ going out}$ .