(c) Let \(\mathbf{A}=\left[\begin{array}{ccc}1 & 1 & 1 \\ 2 & c & 0 \\ -2 & 1 & c\end{array}\right]\), where \(c\) is a real constant.
(i) Use the adjoint method to find \(\mathbf{A}^{-1}\).
(ii) \(\underline{\text { WITHOUT }}\) computing adj \(\left(\mathbf{A}^{\mathrm{T}}\right)\) or \((R+2) \operatorname{adj}\left(\mathbf{A}^{\mathrm{T}}\right)\), find \(\operatorname{det}\left((R+2) \operatorname{adj}\left(\mathbf{A}^{\mathrm{T}}\right)\right)\).
(Note: The answers of (c)(i) and (ii) are in terms of \(c\).)
3. (3pts) Consider the \(3 \times 3\) matrices \(B=\left[\begin{array}{ccc}1 & 1 & 2 \\ -1 & 0 & 4 \\ 0 & 0 & 1\end{array}\right]\) and \(A=\left[\begin{array}{lll}\mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3}\end{array}\right]\), where \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\), and \(\mathrm{a}_{9}\) are the columns of \(A\). Let \(A B=\left[\begin{array}{lll}v_{1} & v_{2} & v_{3}\end{array}\right]\), where \(v_{1}, v_{2}\), and \(v_{3}\) are the columns of the product. Express a as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\), and \(\mathbf{v}_{3}\).4. (3pts) Let \(T(x)=A x\) be the linear transformation given by$$...
I have the first method complete, but I can't figure out the second method Could someone please show how to use the second method?2. Find the unit step response of:$$ \begin{aligned} \dot{\overrightarrow{\mathbf{x}}}(t) &=\left[\begin{array}{cc} 0 & 1 \\ -2 & -2 \end{array}\right] \overrightarrow{\mathbf{x}}(t)+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] u(t) \\ y(t) &=\left[\begin{array}{cc} 2 & 3 \end{array}\right] \overrightarrow{\mathbf{x}}(t) \end{aligned} $$by two methods (1): transfer function and then (2) \(y(t)=\mathbf{C} e^{\mathbf{A} t} \overrightarrow{\mathbf{x}}(0)+\mathbf{C} \int_{0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} u(\tau) d \tau+\mathbf{D} u(t)\). Re-member that the Laplace...
A certain FM signal is given by: \(s(t)=A_{c} \cos \left[2 \pi f_{c} t+2 \pi k_{f} \int_{0}^{t} m(\tau) d \tau\right]\). Estimate the bandwidth of \(s(t)\) using Carson's rule. Assume the spectrum of m(t) is given by:$$ \begin{array}{rr} M(f)=\operatorname{rect}(f) \text { and } \operatorname{rect}(f)=1 & -10 \mathrm{kHz} \leq f \leq 10 \mathrm{kHz} \\ \operatorname{rect}(f)=0 & |f| \geq 10 \mathrm{kHz} \end{array} $$Assume \(k_{f}=2 \mathrm{kHz} /\) volt. Also estimate the bandwidth using Figure 4.9. Compare your results.(Hint: Consider your response to question 5.) from...
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
While using a pencil sharpener, a student applies the forces and couple shown.(a) Determine the forces exerted at \(\mathrm{B}\) and \(\mathrm{C}\) knowing that these forces and the couple are equivalent to \(\mathrm{a}\)force couple system at A consisting of the force \(\mathbf{R}=(2.6 \mathrm{lb}) \mathrm{i}+R_{yj} -(0.7 \mathrm{lb}) \mathbf{k}\) and the couple \(M_{A}^{R}=M_{x} \mathbf{i}+\)\(\left(1.0 lb* f t\right) j-\left(0.72 l b* f t\right) \mathbf{k} .\)(b) Find the corresponding values of \(R_{y}\) and \(M_{x}\).
This problem uses least squares to find the curve \(y=a x+b x^{2}\) that best fits these 4 points in the plane:$$ \left(x_{1}, y_{1}\right)=(-2,2), \quad\left(x_{2}, y_{2}\right)=(-1,1), \quad\left(x_{1}, y_{3}\right)=(1,0), \quad\left(x_{4}, y_{4}\right)=(2,2) . $$a. Write down 4 equations \(a x_{i}+b x_{i}^{2}=y_{i}, i=1,2,3,4\), that would be true if the line actually went through a11 four points.b. Now write those four equations in the form \(\mathbf{A}\left[\begin{array}{l}a \\ b\end{array}\right]=\mathbf{y}\)c. Now find \(\left[\begin{array}{l}\hat{a} \\ \hat{b}\end{array}\right]\) that minimizes \(\left\|A\left[\begin{array}{l}a \\ b\end{array}\right]-\mathbf{y}\right\|^{2}\).
In an article on measuring flows from pipes, the author calculated a volumetric flow rate, \(q=80.8\left(\mathrm{~m}^{3} / \mathrm{s}\right)\) using the formula:\(q=C A_{1} \sqrt{\frac{2 g V\left(p_{1}-p_{2}\right)}{\left(1-\left(\frac{A_{1}}{A_{2}}\right)^{2}\right)}}\)\(\begin{array}{ccc}q & \text { volumetric flow rate } & m^{3} / s \\ C & \text { dimensionless coefficient } & 0.6 \\ A_{1} & \text { cross-sectional area } & 1.0 \mathrm{~m}^{2} \\ A_{2} & \text { cross-sectional area } & 2.0 \mathrm{~m}^{2} \\ V & \text { specific volume } & 0.001 \mathrm{~m}^{3} /...
Problem \(1 \quad\) Bivariate normal distributionAssume that \(\boldsymbol{X}\) is a bivariate normal random variable with$$ \boldsymbol{\mu}=E \boldsymbol{X}=\left(\begin{array}{l} 0 \\ 2 \end{array}\right) \quad \text { and } \quad \Sigma=\operatorname{Cov} \boldsymbol{X}=\left(\begin{array}{ll} 3 & 1 \\ 1 & 3 \end{array}\right) $$Let$$ \boldsymbol{Y}=\left(\begin{array}{l} Y_{1} \\ Y_{2} \end{array}\right)=\left(\begin{array}{lr} 1 / \sqrt{2} & -1 / \sqrt{2} \\ 1 / \sqrt{2} & 1 / \sqrt{2} \end{array}\right) \boldsymbol{X} $$a) Find the mean vector and covariance matrix of \(Y\). What is the distribution of \(Y ?\) Are \(Y_{1}\) and...
Suppose that \(\left(\xi_{j}\right)^{\infty}=1\) is a sequence of independent identically distributed \((i . i . d .)\) continuous random variables.- Suppose that each \(\xi_{i}\) has a probability density function \(p_{i}(x)=\left\{\begin{array}{c}\frac{\beta}{x^{x}}, x \geq 1 \\ 0, x<1\end{array}\right.\), where \(\alpha, \beta \in\)R.- Let \(S_{n}=\sum_{i=1}^{n} \xi_{i}\).- Let \(S_{n}=\frac{s_{n}-\operatorname{mE}\left(\xi_{0}\right)}{\sqrt{n} \operatorname{Var}(\mathcal{B})}\).a. Find a condition on \(\alpha\) and a condition on \(\beta\) (as a function of \(\alpha\) ) which together make \(f_{1}(x)\) a probability density function.b. Find conditions on \(\alpha\) which guarantee that \(\lim _{n \rightarrow \infty}...
Let \(T: R^{3} \rightarrow R^{2}\) defined by \(T\left(\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]\right)=\left[\begin{array}{c}2 x_{1}+x_{3} \\ -x_{2}\end{array}\right]\).a. Find the matrix \(A\) such that \(T(x)=A x\)b. Demonstrate that \(T\) is a linear transformation.