Multiplying Matrices (Grades 11-12)

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Multiplying Matrices

1. 
For given matrices A and B, let [math]A \ B = M[/math], where M is a also a matrix. Which of the following correctly describes the dimensions of matrix M?
  1. Number of rows of A, number of columns of B.
  2. Number of rows of B, number of columns of A.
  3. Number of rows of A, number of columns of A.
  4. Number of rows of B, number of columns of B.
2. 
Multiply the matrices, if possible. [math] [[3,4],[-2,5],[0,2]] * [[4,-1],[5,3]] [/math]
  1. [math] [[8,-13,-2],[27,5,6]][/math]
  2. [math] [[12,-4],[-10,15],[0,2]] [/math]
  3. [math] [[32,9],[17,17],[10,6]][/math]
  4. Not possible.
3. 
Evaluate. [math] [[4,8,-1,-1,3]] * [[2],[0],[-1],[3],[4]] [/math]
  1. [math] [18] [/math]
  2. [math] [[8],[0],[1],[-3],[12]] [/math]
  3. [math] [[8,0,1,-3,12]][/math]
  4. These matrices cannot be multiplied together.
4. 
Multiply, if possible. [math] [[4,0],[-3,6]] * [[3,-4],[5,1]] [/math]
  1. [math] [[12,0],[-15,6]][/math]
  2. [math] [[12,20],[-33,-9]] [/math]
  3. [math] [[12,-16],[21,18]] [/math]
  4. Not possible.
5. 
Multiply. [math] [[2,-1],[0,7]] * [[-1,-4],[6,1]] [/math]
  1. [math] [[-8,-9],[42,7]] [/math]
  2. [math] [[-2,4],[0,7]] [/math]
  3. [math] [[1,-5],[6,8]] [/math]
  4. [math] [[-8,-27],[12,1]] [/math]
6. 
Find the product, if possible. [math] [[4,5],[-1,0],[9,3]] * [[-1,-2],[5,4],[0,3]] [/math]
  1. [math] [[-4,-10],[-5,0],[0,9]] [/math]
  2. [math] [[-9,15],[-5,-1]][/math]
  3. [math] [[-14],[-5],[9]] [/math]
  4. Not possible.
7. 
Multiply, if possible. [math] [[3,2,-1],[0,9,1],[3,2,-5]] * [[1,0,1],[-2,-3,2],[5,6,-1]] [/math]
  1. [math] [[3,0,-1],[0,-27,2],[15,12,5]] [/math]
  2. [math][[2,-14,28],[1,-25,53],[-2,-22,32]] [/math]
  3. [math][[-5,-12,9],[-23,-22,18],[-25,-37,16]][/math]
  4. [math] [[-6,-12,8],[-13,-21,17],[-26,-36,12]][/math]
8. 
Shane has a matrix of three ordered pairs, [math]P = [[3,2],[8,-1],[5,0]][/math], where the x-coordinates are in the first column and the y-coordinates are in the second column. He wants to find a matrix A, such that the product [math]A \ P[/math] gives a matrix that will only have the first and third ordered pairs. Which of the following matrices will give this result?
  1. [math]A = [[1,1],[0,0],[1,1]][/math]
  2. [math]A = [[1,0,0],[0,0,1]][/math]
  3. [math]A = [[1,0,1]][/math]
  4. [math]A = [[1,0,1],[1,0,1]] [/math]
9. 
Given the matrix multiplication equation [math]A \ B = C[/math], where [math]C = [[4,3,2,0],[-1,3,5,9]][/math], which of the following are possible matrices for B? There may be more than one correct choice.
  1. [math]B = [[1,0,0,0],[0,0,0,1]][/math]
  2. [math]B = [[13,-6,14,14],[0,-6,8,3],[-10,12,1,0],[-3,-6,4,11],[2,15,13,8]][/math]
  3. [math] B = [[3,-2,1,0],[9,3,-11,13],[2,4,-5,-6],[-1,-2,10,6]][/math]
  4. [math] B = [[4,-1],[2,-3],[-2,1],[0,5]][/math]
10. 
Kathi recently did a survey, asking which grocery store people shopped at most frequently. There were 5 possible grocery stores to choose from. She asked 3 different groups of people and then put this data in a matrix, each row representing a different group of people, each column representing a different grocery store. Let this be matrix G. Kathi wants to find a matrix, A, such that [math]G \ A = S[/math], where the matrix S will be the sum of each row (or the total number of people in each group). Which of the following would correctly accomplish this?
  1. [math]A = [[1,1,1,1,1]][/math]
  2. [math]A = [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]][/math]
  3. [math]A = [[1],[1],[1],[1],[1]][/math]
  4. Without knowing the entries of matrix G, this cannot be determined.

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