1. Which of the following class topics relate to determinants?
   a) Invertibility of a 2x2 matrix
   b) A determinant 1 (or -1) coding matrix with integer entries will ensure that we don't pick up fractions in the decoding matrix.
   c) Both a) and b)
   d) Neither a) nor b)
   
   
   
   
   
2. Which of the following matrices does not have an inverse?
   a) Matrix([[1,2],[3,4]])
   b) Matrix([[2,2],[4,4]])
   c) Matrix([[0,4],[2,0]])
   d) More than one of the above do not have inverses
   e) All of the above have inverses
   
   
   
   
   
   
3. Which of the following are true about the matrix A=Matrix([[1,0],[k,1]])
   a) Determinant of A is 1
   b) A is a vertical shear matrix
   c) When we perform AB_2xn then we have applied r₂'=k r₁ + r₂ to B, because it is the elementary matrix representing that row operation
   d) More than one of the above
   e) All of a), b) and c)
   
   
   
   
   
   
4. Which of the following statements is true?
   (a) If a square matrix has two identical rows then its determinant is zero.
   (b) If the determinant of a matrix is zero, then the matrix has two identical rows.
   (c) Both are true.
   (d) Neither is true.
   
   
   
   
5. Suppose the determinant of matrix A is zero. How many solutions does the system Ax = 0 have?
   a) 0
   b) 1
   c) 2
   d) infinite
   e) other
   
   
   
   
6. We find that for a square coefficient matrix A, the homogeneous matrix equation Ax =0, has only the trivial solution x=0. This means that
   (a) Matrix A has a 0 determinant.
   (b) Matrix A has a non-zero determinant.
   (c) This tells us nothing about the determinant.
   
   
   
   
7. Suppose the determinant of matrix A is zero. How many solutions does the system Ax = b have?
   a) 0
   b) 1
   c) Infinite
   d) 0, 1, or infinite---it depends on what b is.
   e) 0 or infinite---it depends on what b is.
   
   
   
   
8. If A is an invertible matrix, what else must be true?
   (a) If AB=C then B=A^-1C
   (b) the columns of A span the entire space
   (c) 5A is invertible
   (d) The reduced row echelon form of A is I
   (e) All of the above must be true
   
   
   
   
9. In exercise 3.3 #19, the area of the parallelogram is 8, because that is the determinant of A=Matrix([[5,6],[2,4]]). Can we find a rectangle that creates a matrix that is row equivalent to A with the same area?
   a) Impossible with the conditions given
   b) It is possible but I am stuck on how to do so
   c) Yes and I can explain how
   
   
   
   
10. By hand, use Laplace expansion as directed:
    Step 1: First expand down the first column to take advantage of the 0s. You'll have one nonzero term.
    Step 2: then down the 1st column of the next 4x4 matrix
    Step 3: then along the 3rd row of the 3x3 matrix:
    
    The determinant is
    (a) 100
    (b) 0
    (c) -100
    (d) -10
    (e) None of the above.
    
    
    
    
    Solutions
    1. c)
    2. b)
    3. e)
    4. a)
    5. d)
    6. b)
    7. e)
    8. e)
    9. c)
    10. c)