Using Matrices to Represent and Manipulate Data (Grades 11-12)

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Using Matrices to Represent and Manipulate Data

Jack is keeping track of the scores for his favorite teams in a series of basketball games. He records the initials of the team and score for each game. Which matrix represents the data he collected?
Round 1
SC 67
RG 103
PD 89

Round 2
RG 109
SC 86
PD 111

Round 3
PD 42
SC 99
RG 121
  1. [math][[67,103,89],[109,86,111],[42,99,121]][/math]
  2. [math][[67,86,99],[103,109,121],[89,111,42]][/math]
  3. [math][[42,99,121],[111,86,109],[67,103,89]][/math]
  4. [math][[67,109,42],[103,86,99],[89,111,122]][/math]
Clayton is recording information for a science experiment. He has three separate plants in different areas, and is recording their height each week. So far he has recorded data for four weeks, and each plant always increased in height. The heights of the first plant are 3 cm, 6 cm, 8 cm, 12 cm. The heights of the second plant are 2 cm, 4 cm, 5 cm, 7 cm. The heights of the third plant are 2 cm, 5 cm, 8 cm, 10 cm. Which of the following matrices correctly represent this data? Choose all that apply.
  1. [math][[3,6,8,12],[2,4,5,7],[2,5,8,10]][/math]
  2. [math] [[3,2,2],[6,4,5],[8,5,8],[12,7,10]] [/math]
  3. [math] [[2,2,3],[4,5,5],[6,7,8],[8,10,12]][/math]
  4. [math] [[12,8,6,3],[7,5,4,2],[2,5,8,10]][/math]
Which of the following matrix equations correctly represents this system of equations?
[math]4x + 3y - z = 12, \ \ 3y+x+z= 10, \ \ 8x - z - 3y = 1[/math]
  1. [math][[4,3,-1],[3,1,1],[8,-1,-3]] [[x],[y],[z]] = [[12],[10],[1]] [/math]
  2. [math][[4,3,-1],[3,1,1],[8,-1,-3]] [[12],[10],[1]] = [[x],[y],[z]] [/math]
  3. [math][[4,3,-1],[1,3,1],[8,-3,-1]] [[x],[y],[z]] = [[12],[10],[1]] [/math]
  4. [math][[4,1,8],[3,3,-3],[-1,1,-1]] [[x],[y],[z]] = [[12],[10],[1]] [/math]
Matrices can be used to send coded messages. One way this is done is by taking a message, changing the letters into numbers, and then putting these numbers into a matrix. Zero represents a blank space. The matrix is formed by starting with the upper left most element, and going down each column till its full, then starting on the next column. If there are extra spaces in a matrix, fill the remaining elements with zeros (for example, if a message only has 7 letters, and one uses a 3-by-3 matrix, the remaining 2 elements would be zeros). Call the matrix containing the message matrix M. Then, one multiplies this by an encoding matrix; call this matrix A. Thus, the message sent would be the matrix AM. The recipient knows the encoding matrix, A. All they have to do is to multiply by the inverse, [math]A^{-1}[/math].

For the coded message [math][[24,20,56,76,55,24,20,56,37],[87,78,206,267,185,87,78,206,127],[27,95,173,118,-110,27,95,173,1]][/math], and the coding matrix [math][[-1,3,4],[-3,10,14],[2,-6,7]][/math], what is the message?
Andrea has been collecting data on typical house prices in three neighborhoods. She has data for 2016, 2017, and 2018. She puts this data into a matrix, labeled H, where each column represents a different neighborhood and each row represents a certain year (2016 for the first row, etc.)

[math]H = [["812,000", "616,000", "1,212,000"],["856,000", "583,000", "1,108,000"],["881,000", "612,000", "1,190,000"]] [/math]

If she wanted to find the average of the house prices over the three years for each neighborhood, which of the following operations would yield that information in a matrix?
  1. [math]1/3 [[1, 1, 1]]H[/math]
  2. [math]1/3 [[1],[1],[1]] H[/math]
  3. [math]1/3 H [[1, 1, 1]] [/math]
  4. [math]1/3 H [[1],[1],[1]] [/math]
Kiera has a few apple trees, peach trees, and pear trees. She has kept track of the total volume of fruit, in bushels, each fruit tree has produced for the years 2016, 2017, and 2018. She has put this in a matrix, where each type of fruit tree is in a different column (apples, peaches, and pears, respectively), and each row is a certain year, starting with 2016 as the first row. Let this be matrix F.

[math]F =[[45, 6, 12],[40, 3, 10],[50, 5, 15]] [/math]

Which operation would give the total volume of fruit produced, in bushels, for each year?
  1. [math][[1,1,1]] F[/math]
  2. [math][[1],[1],[1]] F[/math]
  3. [math]F [[1,1,1]][/math]
  4. [math]F [[1],[1],[1]][/math]
Matrices are widely used in image processing, where images are taken and manipulated in some way. To simplify this, consider a grayscale image (an image without color, but with varying hues of gray). Each pixel in the image has an assigned value, from 0 (completely black) to 255 (completely white). Usually, an image would be comprised of thousands or more pixels, but again, for simplification, consider a 3-by-3 matrix. Let the image be represented by the matrix A, as follows.

[math]A = [[124, 127, 129], [118, 120, 124], [115, 119, 121]][/math]

Which of the following operations would "invert" the grayscale image- that is, give the opposite intensity of each pixel? For example, if the original image has a pixel that was almost white, 4, the same pixel in the resulting image would be 251, almost black.
  1. [math]A^{-1}[/math] (inverse of matrix A)
  2. [math]A^T[/math] (transpose of matrix A)
  3. [math]255*[[1,1,1],[1,1,1],[1,1,1]] - A[/math]
  4. [math]A - 255*[[1,1,1],[1,1,1],[1,1,1]][/math]
The following matrix, A, lists 5 sets of points, where the first column is the x coordinate and the second column is the y coordinate.

[math]A = [[3,2],[4,1],[5,9],[7,-1],[1,8]] [/math]

The following questions will use matrix operations to find the distance between the first and third points.
Which of the following operations would result in a 2-by-2 matrix, with only the first and third points as elements? Let this matrix be called B.
  1. [math]B = [[1,0,1,0,0],[1,0,1,0,0]] A[/math]
  2. [math]B = A [[1,1],[0,0],[1,1],[0,0],[0,0]][/math]
  3. [math]B = A [[1,0],[0,0],[0,1],[0,0],[0,0]][/math]
  4. [math]B = [[1,0,0,0,0],[0,0,1,0,0]] A[/math]
Which of the following operations would create a matrix, called C, where each element is the difference of the respective coordinates of the two points? I.e., the first element is the difference of the two x-coordinates.
  1. [math]C = B[[1,-1]][/math]
  2. [math]C = [[1,-1]] B[/math]
  3. [math]C = [[1],[-1]] B[/math]
  4. [math]C = B [[1],[-1]][/math]
The transpose of a matrix is when the rows and columns are switched. For a given matrix, [math]M[/math], and its transpose, [math]M^T[/math], the columns of [math]M[/math] correspond to the rows of [math]M^T[/math]. This means that the dimensions of the matrix are switched; a 2-by-4 matrix becomes a 4-by-2 matrix. For a 1-by-n or m-by-1 matrix, the single column becomes a single row (or vice-versa).

Which of the following operations would result in a matrix of only one element which is the square of the distance between the first and third points in the original matrix A? (Taking the square root of this element would therefore give the distance between these points).
  1. [math]C^T C[/math]
  2. [math]C C^T[/math]
  3. [math](C^T C)^2[/math]
  4. [math](C C^T)^2[/math]

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