Find the characteristic polynomial of each of the following matrices:

$$\begin{align*} & (a)\begin{bmatrix} 2 & 1\\ -1 & 3 \end{bmatrix} & & (b) \begin{bmatrix} 1 & 2 & 1\\ 0 & 1 & 2\\ -1 & 3 & 2 \end{bmatrix} \\\ \\ & (c) \begin{bmatrix} 4 & -1 & 3\\ 0 & 2 & 1\\ 0 & 0 & 3 \end{bmatrix} & & (d) \begin{bmatrix} 4 & 2 \\ 3 & 3 \end{bmatrix} \end{align*}$$

Compute

$$\mathbf { u } \cdot \mathbf { V }$$

where: (a)

$$\mathbf { u } = \left[ \begin{array} { l } { 2 } \\ { - 1 } \\ { 3 } \end{array} \right] , \mathbf { v } = \left[ \begin{array} { r } { - 1 } \\ { 1 } \\ { 1 } \end{array} \right]$$

, (b)

$$\mathbf { u } = \left[ \begin{array} { r } { 1 } \\ { 2 } \\ { - 1 } \end{array} \right] , \mathbf { v } = \mathbf { u }$$

, (c)

$$\mathbf { u } = \left[ \begin{array} { r } { 1 } \\ { 1 } \\ { - 3 } \end{array} \right] , \mathbf { v } = \left[ \begin{array} { r } { 2 } \\ { - 1 } \\ { 1 } \end{array} \right]$$

, (d)

$$\mathbf { u } = \left[ \begin{array} { r } { 3 } \\ { - 1 } \\ { 5 } \end{array} \right] , \mathbf { v } = \left[ \begin{array} { r } { 6 } \\ { - 7 } \\ { - 5 } \end{array} \right]$$

, (e)

$$\mathbf { u } = \left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] , \mathbf { v } = \left[ \begin{array} { l } { a } \\ { b } \\ { c } \end{array} \right]$$

, (f)

$$\mathbf { u } = \left[ \begin{array} { l } { a } \\ { b } \\ { c } \end{array} \right] , \mathbf { v } = \mathbf { 0 }$$

Find the standard matrix of the linear transformation T, and use it to determine whether T is onto. $T:R^2\to R^3$ defined by

$$\begin{equation*}T\begin{pmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}\end{pmatrix}= \begin{bmatrix}3x_1-x_2\\x_2\\x_1+x_2\end{bmatrix}.\end{equation*}$$

Compute $A^T+2B^T$ if

$$\begin{equation*}A=\begin{bmatrix}2&&-1&&5\\ 3&&4&&1 \end{bmatrix}, B=\begin{bmatrix}1&&0&&-2\\ 2&&3&&4 \end{bmatrix}\end{equation*}$$

(I) How much work is required to stop an electron $\left(m=9.11 \times 10^{-31} \mathrm{~kg}\right)$ which is moving with a speed of $1.10 \times 10^6 \mathrm{~m} / \mathrm{s}$?

Find a basis for $S^\perp$.

$$S=\begin{Bmatrix}\begin{bmatrix}2\\-1\\3\end{bmatrix}\end{Bmatrix}$$

$$\begin{equation*}A=\begin{bmatrix}a&&b\\c&&d\end{bmatrix}, E=\begin{bmatrix}1&&0\\0&&k\end{bmatrix}\end{equation*}$$

Verify that $\text{det}EA=(\text{det}E)(\text{det}A)$.

Given $m \in \mathbb{Z}$, we say that rn is even if $m / 2 \in \mathbb{Z}$, and m is odd otherwise. (a) Show that if m is odd, m = 2n + 1 for some $n \in \mathbb{Z}$. (b) Show that if p and q are odd, so are $p \cdot q$ and $p^n$, or any $n \in \mathbb{Z}_{+}$. (c) Show that if a > 0 is rational, then a = m/n for some m, where $n \in \mathbb{Z}_{+}$ not both m and n are even. (d) Theorem $\sqrt{2}$ is irrational.

A matrix and a vector are given. Show that the vector is an eigenvector of the matrix and determine the corresponding eigenvalue.

$$\begin{equation*}\begin{bmatrix}19&&-7\\42&&-16\end{bmatrix}, \begin{bmatrix}1\\3\end{bmatrix}.\end{equation*}$$

A une soirée, vous jouez a la voyante (fortune-teller) et prédisez la carriere de chacun(e) de vos ami(e)s. Choisissez le verbe convenable pour décrire vos prévisions.

Verbes: écrire, enseigner, jouer, participer, s'occuper, travailler, vendre, voyager

Vous dans une ecole primaire a Seattle.

$$\begin{equation*}A=\begin{bmatrix}2&&1\\4&&-1\end{bmatrix}, B=\begin{bmatrix}2&&3\\4&&6\end{bmatrix}\end{equation*}$$

Compute $A^{-1}B^T$, or give the reason why the expression is not defined.

(a) Calculate the sum $S_n$ for n=1, 2, 3, ..., 10, where if n<r, then

$$\left(\begin{array}{l}n \\ r\end{array}\right)=0$$

.

(b) Predict a general formula for $S_n$.

$$\left(\begin{array}{l}n \\ 1\end{array}\right)+\left(\begin{array}{l}n \\ 3\end{array}\right)+\left(\begin{array}{l}n \\ 5\end{array}\right)+\left(\begin{array}{l}n \\ 7\end{array}\right)+\cdots$$

Find AB and BA, if possible.

$$\begin{aligned} &A=\left[\begin{array}{l} 4 \\ 5 \end{array}\right]\\ &B=\left[\begin{array}{llll} 6 & 1 & -10 & 9 \end{array}\right] \end{aligned}$$

Draw two lines under the simple predicate in given sentence. Then write the tense of the verb in the space provided.
_____Women had not received the right to vote yet.