Find $(a)$ $A^TA$ and $(b)$ $AA^T$. Show that each of these products is symmetric.

$A=$

$$\begin{bmatrix} 4 & -3 & 2 & 0 \\ 2 & 0 & 11 &-1 \\ -1 & -2 & 0 & 3 \\ 14 & -2 & 12 & -9 \\ 6 &8 & -5 &4 \end{bmatrix}$$

Define $T: P_{3} \rightarrow P_{3}$ by $T(p)=p(x)+p^{\prime}(x)$ Find the rank and nullity of T.

When a quantity of hot water is mixed with a quantity of cold water, the combined system comes to thermal equilibrium at some intermediate temperature. How does the entropy of the system (both liquids) change?

Use an inverse matrix to solve each system of linear equations or matrix equation.

$$\begin{array}{l} {-x_{1}+x_{2}+2 x_{3}=1} \\ {2 x_{1}+3 x_{2}+x_{3}=-2} \\ {5 x_{1}+4 x_{2}+2 x_{3}=4} \end{array}$$

The augmented matrix represents a system of linear equations that has been reduced using Gauss-Jordan elimination. Write a system of equations with nonzero coefficients that the reduced matrix could represent.

$$\begin{bmatrix} 1&0&3&-2 \\ 0&1&4&1 \\ 0&0&0&0 \end{bmatrix}$$

There are many correct answers.

A hotel chain charges $75 each night for the first two nights and$50 for each additional night’s stay. Express the total cost T as a function of the number of nights x that a guest stays.

Let u be an n $\times$ 1 column matrix satisfying u$^T$u = I$_1$. The n $\times$ n matrix H = I$_n$ − 2uu$^T$ is called a Householder matrix.

Prove that H is symmetric and nonsingular.

Find the inverse of the matrix (if it exists).

$$A=\begin{bmatrix} 0.6& 0 & -0.3\\ 0.7 & -1& 0.2 \\ 1 & 0 & -0.9 \end{bmatrix}$$

Prove that a rotation of $\theta$, where $\cot 2 \theta=(a-c) / b$, will eliminate the xy-term from the equation $a x^{2}+b x y+c y^{2}+d x+e y+f=0$.

Perform the operations when

$A=$

$$\begin{bmatrix} 1 & 2 \\ 0 & -1 \\ \end{bmatrix}$$

.

Evaluate

$$AI$$

A corporation has three factories, each of which manufactures acoustic guitars and electric guitars. In the matrix

$$\begin{bmatrix} 70 & 35 & 50\\ 100 & 25 & 75 \end{bmatrix}$$

$a_{ij}$ represents the number of guitars of type $i$ produced at factory $j$ in one day. Find the production levels when production increases by $20\%$

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

$A=$

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

$B=$

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

Solve the system of first-order linear differential equations.

$$\begin{aligned} &y_{1}^{\prime}=y_{1}-y_{2}\\ &y_{2}^{\prime}=2 y_{1}+4 y_{2} \end{aligned}$$

In example showed that the $\beta$ decay of $^{4} \mathrm{Be}^{7} \text { to }^{3} \mathrm{Li}^{7}$ proceeds only through electron capture because the atomic mass difference is 0.00093u, which is less than two electron rest masses. Consider a $^{4} \mathrm{Be}^{7}$ nucleus, initially at rest, that captures a K electron and emits a neutrino. a) Estimate the recoil velocity of the nucleus after the process is completed. (Hint: The recoil energy of the nucleus is negligibly small.) b) Suggest a technique for detecting electron capture.