The Zero and Identity Matrices (Grades 11-12)

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The Zero and Identity Matrices

1. 
The zero matrix is always square.
  1. True
  2. False
2. 
The identity matrix is always square.
  1. True
  2. False
3. 
If [math]I[/math] is the identity matrix, and [math]A[/math] has the same dimensions as [math]I[/math], then [math]AI = IA[/math].
  1. True
  2. False
4. 
For a non-zero matrix A, if [math]AB=0[/math], then B must be the zero matrix.
  1. True
  2. False
5. 
For the 3-by-3 identity matrix [math]I_3[/math], and a given non-square matrix [math]A[/math], if [math]I_3A = A[/math], then [math]AI_3=A[/math].
  1. True
  2. False
6. 
Which of the following are identity matrices? There may be more than one correct answer.
  1. [math][[1,0,0],[0,1,0],[0,0,1]][/math]
  2. [math][[1,0,0],[1,0,0],[1,0,0]][/math]
  3. [math][[1,0],[0,1]][/math]
  4. [math][[1,1],[1,1]][/math]
7. 
If [math]A[/math] is a 5-by-3 non-zero matrix, which of the following expressions, if evaluated, would yield the matrix [math]A[/math] as the answer? Choose all correct answers.
  1. [math]A + [[0,0,0],[0,0,0],[0,0,0],[0,0,0],[0,0,0]] [/math]
  2. [math] [[0,0,0],[0,0,0],[0,0,0],[0,0,0],[0,0,0]] + A[/math]
  3. [math]A + [[0,0,0],[0,0,0],[0,0,0]] [/math]
  4. [math]A + [[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0]] [/math]
8. 
If [math]B[/math] is a 3-by-2 matrix, which of the following expressions, if evaluated, would yield the matrix [math]B[/math] as the answer? Choose all correct answers.
  1. [math]B * [[1,0,0],[0,1,0],[0,0,1]][/math]
  2. [math]B * [[1,0],[0,1]][/math]
  3. [math] [[ 1,0,0],[0,1,0],[0,0,1]] * B[/math]
  4. [math][[1,0],[0,1]] * B[/math]
9. 
Given a matrix A, is there always another matrix, B, such that their sum is the zero matrix? What are the elements of the matrix B?
  1. No, there isn't always such a matrix B. If there is, its elements are the negative of the corresponding elements in matrix A.
  2. No, there isn't always such a matrix B. If there is, its elements are determined on a case-by-case basis, with no general rule.
  3. Yes, there is always such a matrix. The elements of B are found on a case-by-case basis, with no general rule.
  4. Yes, there is always such a matrix. The elements of B are the negative of the corresponding elements in A.
10. 
Adrian was learning about the identity matrix and multiplicative inverses in math class, and is now working on his homework. He knows that, for a square matrix [math]A[/math], he can usually find another matrix [math]B[/math] such that [math]AB = I[/math], where [math]I[/math] is the identity matrix of the same size as [math]A[/math] and [math]B[/math]. Adrian is now considering the matrix [math]A = [[3,2],[6,4]][/math]. The way he has been finding the multiplicative inverse matrix is to create another matrix [math]B = [[a,b],[c,d]][/math]. He performs this multiplication, and it yields two systems of equations, as follows.

[math]{{:(3a + 2c , = 1),(6a + 4c, = 0):}[/math]

[math]{{:(3b + 2d , = 0),(6b + 4d, = 1):}[/math]

Adrian realizes that both these systems are inconsistent. Is Adrian's work correct? And if so, what does this imply?
  1. No, Adrian's work is not correct. His entire method for finding multiplicative inverses is wrong.
  2. No, Adrian's work is not correct. Although his general method is OK, he has made a mistake when multiplying the matrices.
  3. Yes, Adrian's work is correct. This means that, in this case, no such B matrix exists, and therefore not all square matrices have a multiplicative inverse.
  4. Yes, Adrian's work is correct. This implies that, in this case, the B matrix must be the zero matrix.

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