# Let B be a 4 times 4 matrix to which we apply the following operations: 1. double column 1, 2. halve row 3, 3. add row 3 to row 1, 4. interchange columns 1 and 4, 5. subtract row 2 from each of the other rows, 6. replace column 4 by column 3, 7. delete column 1 (so that the column dimension is reduced by 1). (a) Write the result as a product of eight matrices. (b) Write it again as a product ABC (same B) of three matrices.

Question
Matrices
Let B be a $$4 \times 4$$ matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (so that the column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product ABC (same B) of three matrices.

2021-02-26
Step 1
Note: Here multiple questions posted, answer to the first three subparts are given. If there is a need for remaining too, kindly post it again with the remark.
Let B be a $$4 \times 4$$ matrix,
Let, $$B=\begin{bmatrix}b_{11} & b_{12}&b_{13}&b_{14} \\ b_{21} & b_{22}&b_{23}&b_{24} \\ b_{31} & b_{32}&b_{33}&b_{34} \\ b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}$$
where, ($$b_{ij}$$) represents the element at the position, i'th row, and j'th column. $$R_i$$ represents i'th row and $$C_j$$ represents j'th column.
Step 2
1) Double column 1:
$$C_1 \rightarrow 2C_1$$
$$B\sim\begin{bmatrix}2b_{11} & b_{12}&b_{13}&b_{14} \\ 2b_{21} & b_{22}&b_{23}&b_{24} \\ 2b_{31} & b_{32}&b_{33}&b_{34} \\ 2b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}$$
2) Halve row 3:
$$R_3 \rightarrow \frac{1}{2}R_3$$
$$B=\begin{bmatrix}b_{11} & b_{12}&b_{13}&b_{14} \\ b_{21} & b_{22}&b_{23}&b_{24} \\ \frac{1}{2}b_{31} & \frac{1}{2}b_{32}&\frac{1}{2}b_{33}&\frac{1}{2}b_{34} \\\ b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}$$
Step 3
3) Add row 3 to row 1:
$$R_1 \rightarrow R_1+R_3$$
$$B=\begin{bmatrix}b_{11}+b_{31} & b_{12}+b_{32}&b_{13}+b_{33}&b_{14}+b_{34} \\ b_{21} & b_{22}&b_{23}&b_{24} \\ b_{31} & b_{32}&b_{33}&b_{34} \\\ b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}$$

### Relevant Questions

Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.
In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung.
1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?
2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
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6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
Is statistical inference intuitive to babies? In other words, are babies able to generalize from sample to population? In this study,1 8-month-old infants watched someone draw a sample of five balls from an opaque box. Each sample consisted of four balls of one color (red or white) and one ball of the other color. After observing the sample, the side of the box was lifted so the infants could see all of the balls inside (the population). Some boxes had an “expected” population, with balls in the same color proportions as the sample, while other boxes had an “unexpected” population, with balls in the opposite color proportion from the sample. Babies looked at the unexpected populations for an average of 9.9 seconds (sd = 4.5 seconds) and the expected populations for an average of 7.5 seconds (sd = 4.2 seconds). The sample size in each group was 20, and you may assume the data in each group are reasonably normally distributed. Is this convincing evidence that babies look longer at the unexpected population, suggesting that they make inferences about the population from the sample? Let group 1 and group 2 be the time spent looking at the unexpected and expected populations, respectively. A) Calculate the relevant sample statistic. Enter the exact answer. Sample statistic: _____ B) Calculate the t-statistic. Round your answer to two decimal places. t-statistic = ___________ C) Find the p-value. Round your answer to three decimal places. p-value =
Given the matrix
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}$$
and suppose that we have the following row reduction to its PREF B
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 3&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&0 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}$$
Write $$A \text{ and } A^{-1}$$ as product of elementary matrices.
Let W be the subspace of all diagonal matrices in $$M_{2,2}$$. Find a bais for W. Then give the dimension of W.
If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix
Let $$u=\begin{bmatrix}2 \\ 5 \\ -1 \end{bmatrix} , v=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix} \text{ and } w=\begin{bmatrix}-4 \\ 17 \\ -13 \end{bmatrix}$$ It can be shown that 4u-3v-w=0. Use this fact (and no row operations) to find a solution to the system Ax=b , where
$$A=\begin{bmatrix}2 & -4 \\5 & 17\\-1&-13 \end{bmatrix} , x=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} , b=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix}$$
Let C be the ellipse contained in the xy plane whose equation is $$\displaystyle{4}{x}^{{2}}+{y}^{{2}}={4}$$, oriented clockwise. The force field F described by $$\displaystyle{F}{\left({x},{y},{z}\right)}={x}^{{2}}{i}+{2}{x}{j}+{z}^{{2}}{k}$$, moves a particle along C in the same direction as the curve orientation, performing a W job. C as the surface boundary S: $$\displaystyle{z}={4}-{4}{x}^{{2}}-{y}^{{2}},{z}\ge{0}$$ (with ascending orientation, that is, the component in the z direction equal to 1) and assuming $$\displaystyle\pi={3.14}$$, we can state what:
a) It is not necessary to apply Stokes' Theorem, as C is a closed curve and therefore W = 0.
b) Inverting the orientation of the surface S, we can apply Stokes' Theorem and conclude that W = 12.56.
c) We can apply Stokes' Theorem and conclude that W = 6.28
d) We can apply Stokes' Theorem and conclude that W = 12.56.
The 2 \times 2 matrices A and B below are related to matrix C by the equation: C=3A-2B. Which of the following is matrix C.
$$A=\begin{bmatrix}3 & 5 \\-2 & 1 \end{bmatrix} B=\begin{bmatrix}-4 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-1 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-18 & 5 \\10 & 1 \end{bmatrix}$$
$$\begin{bmatrix}18 & -5 \\-10 & -1 \end{bmatrix}$$
$$\begin{bmatrix}1 & -5 \\-2 & -1 \end{bmatrix}$$
1. The accuracy of the approximation it provides, improves when the trial success proportion p is closer to $$50\%$$