# Matrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if A=begin{bmatrix}2&1&1 -1&-1&4 end{bmatrix} , B=begin{bmatrix}0 & 2 -4 & 12&-3 end{bmatrix} , C=begin{bmatrix}6 & -1 3 & 0-2&5 end{bmatrix} , D=begin{bmatrix}2 & -3&4 -3& 1&-2 end{bmatrix} If the operation is not possible , write NOT POSSIBLE and be able to explain why a)A+B b)B+C c)2A

Question
Matrices
Matrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if
$$A=\begin{bmatrix}2&1&1 \\-1&-1&4 \end{bmatrix} , B=\begin{bmatrix}0 & 2 \\-4 & 1\\2&-3 \end{bmatrix} , C=\begin{bmatrix}6 & -1 \\3 & 0\\-2&5 \end{bmatrix} , D=\begin{bmatrix}2 & -3&4 \\-3& 1&-2 \end{bmatrix}$$
If the operation is not possible , write NOT POSSIBLE and be able to explain why
a)A+B
b)B+C
c)2A

2021-01-05
Step 1
$$\text{Given : } A=\begin{bmatrix}2&1&1 \\-1&-1&4 \end{bmatrix} , B=\begin{bmatrix}0 & 2 \\-4 & 1\\2&-3 \end{bmatrix} , C=\begin{bmatrix}6 & -1 \\3 & 0\\-2&5 \end{bmatrix} , D=\begin{bmatrix}2 & -3&4 \\-3& 1&-2 \end{bmatrix}$$
Solution for question a:
To compute A+B:
Note that two matrices may be added if and only if they have the same dimension, that is, they must have the same number of rows and columns.
Here, Dimension of $$A=2 \times 3$$
And, Dimension of $$B=3 \times 2$$
Matrix A and B do not have the same dimension. Hence, matrix A and B cannot be added.
Therefore it is not possible to perform A+B.
Step 2
Solution for question b:
Here,
Dimension of matrix $$B=3 \times 2$$
Dimension of matrix $$C=3 \times 2$$
Both matrices B and C have the same dimension. Hence, matrix B and C can be added.
Further,
$$B+C=\begin{bmatrix}0 & 2 \\-4 & 1\\2&-3 \end{bmatrix}+\begin{bmatrix}6 & -1 \\3 & 0\\-2&5 \end{bmatrix}$$
$$=\begin{bmatrix}0+6 & 2+(-1) \\-4+3 & 1+0\\2+(-2)&(-3)+5 \end{bmatrix}$$
$$=\begin{bmatrix}6 &1 \\-1 &1\\0&2 \end{bmatrix}$$
Therefore,
$$B+C=\begin{bmatrix}6 &1 \\-1 &1\\0&2 \end{bmatrix}$$
Step 3
Solution for question c:
To compute 2A multiply each entry of the matrix A by 2.
$$2A=2\begin{bmatrix}2&1&1 \\-1&-1&4 \end{bmatrix}$$
$$=\begin{bmatrix}4&2&2 \\-2&-2&8 \end{bmatrix}$$
therefore,
$$2A=\begin{bmatrix}4&2&2 \\-2&-2&8 \end{bmatrix}$$

### Relevant Questions

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Refer to the following matrices.
$$A=\begin{bmatrix}2 & -3&7&-4 \\-11 & 2&6&7 \\6 & 0&2&7 \\5 & 1&5&-8 \end{bmatrix} B=\begin{bmatrix}3 & -1&2 \\0 & 1&4 \\3 & 2&1 \\-1 & 0&8 \end{bmatrix} , C=\begin{bmatrix}1& 0&3 &4&5 \end{bmatrix} , D =\begin{bmatrix}1\\ 3\\-2 \\0 \end{bmatrix}$$
Identify the row matrix. Matrix C is a row matrix.
$$A=\begin{bmatrix}2& 1&1 \\-1 & -1&4 \end{bmatrix} B=\begin{bmatrix}0& 2 \\-4 & 1\\2 & -3 \end{bmatrix} C=\begin{bmatrix}6& -1 \\3 & 0\\-2 & 5 \end{bmatrix} D=\begin{bmatrix}2& -3&4 \\-3 & 1&-2 \end{bmatrix}$$
a)$$A-3D$$
b)$$B+\frac{1}{2}$$
c) $$C+ \frac{1}{2}B$$
(a),(b),(c) need to be solved
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?

Solve for X in the equation, given
$$3X + 2A = B$$
$$A=\begin{bmatrix}-4 & 0 \\1 & -5\\-3&2 \end{bmatrix} \text{ and } B=\begin{bmatrix}1 & 2 \\ -2 & 1 \\ 4&4 \end{bmatrix}$$

The product of matrix B and C is matrix D
$$\begin{bmatrix}2 & -1&4 \\g & 0&3\\2&h&0 \end{bmatrix} \times \begin{bmatrix}-1 & 5 \\4&f\\-3&1 \end{bmatrix}=\begin{bmatrix}i & 24 \\-16&-4\\4&e \end{bmatrix}$$
3.From the expression above, what should be the value of e?
4.From the expression above, what should be the value of g?
5.From the expression above, what should be the value of f?
If $$A=\begin{bmatrix}1 & 1 \\3 & 4 \end{bmatrix} , B=\begin{bmatrix}2 \\1 \end{bmatrix} ,C=\begin{bmatrix}-7 & 1 \\0 & 4 \end{bmatrix},D=\begin{bmatrix}3 & 2 & 1 \end{bmatrix} \text{ and } E=\begin{bmatrix}2 & 3&4 \\1 & 2&-1 \end{bmatrix}$$
Find , if possible,
a) A+B , C-A and D-E b)AB, BA , CA , AC , DA , DB , BD , EB , BE and AE c) 7C , -3D and KE
$$A=\begin{bmatrix}4 & 0 \\-3 & 5 \\ 0 & 1 \end{bmatrix} B=\begin{bmatrix}5 & 1 \\-2 & -2 \end{bmatrix} C=\begin{bmatrix}1 & -1 \\-1 & 1 \end{bmatrix}$$
$$A=\begin{bmatrix}1 & -2&1 \\0 & 2&-1\\2&1&1 \end{bmatrix}$$
$$B=\begin{bmatrix}2 & 1&-1 \\1 & -1&0\\2&-1&1 \end{bmatrix}$$
(a)$$\begin{pmatrix}3 & 2&|&8 \\1 & 5&|&7 \end{pmatrix}$$
(b)$$\begin{pmatrix}5 & -2&1&|&3 \\2 & 3&-4&|&0 \end{pmatrix}$$
(c)$$\begin{pmatrix}2 & 1&4&|&-1 \\4 & -2&3&|&4 \\5 & 2&6&|&-1 \end{pmatrix}$$
(d)$$\begin{pmatrix}4 & -3&1&2&|&4 \\3 & 1&-5&6&|&5 \\1 & 1&2&4&|&8\\5 & 1&3&-2&|&7 \end{pmatrix}$$