1. A_nxn (square). Can Ax=0 have only the trivial solution?
   a) No that statement is impossible
   b) Yes when the columns of A are l.i. but we can't say anything more
   c) Yes when the columns of A are l.i. and A has n pivot rows
   d) Yes when the columns of A are l.i. and A has n pivot columns
   e) Both c and d
   
   
   
   
2. A_mxn (not square). Can Ax=0 have only the trivial solution?
   a) No that statement is impossible
   b) Yes when the columns of A are l.i. but we can't say anything more
   c) Yes when the columns of A are l.i. and A has m pivot rows
   d) Yes when the columns of A are l.i. and A has n pivot columns
   e) Both c and d
   
   
   
   
3. For the Hill Cipher
   a) A_nxn[original message]_nxp=[coded message]_nxp
   b) to decode, we must use apply an invertible matrix to the coded message and read the message along the rows
   c) the method is vulnerable to those that intercept enough coded/decoded vector correspondances because of its linearity
   d) all of the above
   e) more than one of a), b) or c), but not all
   
   
   
   
4. If the condition number of a square matrix with fractional entries is 3.5 10⁶ then
   a) We should use 8 decimal places in our measurements of b if we want solutions to Ax=b to be accurate to 2 decimal places
   b) The matrix is invertible
   c) both of the above
   d) none of the above
   
   
   
   
5. The equation Ax=b has at least one solution for each b in R^n whenever A is an nxn matrix.
   a) true
   b) false and I can think of a counterexample
   c) false and I can think of a correction
   d) both b) and c)
   e) other
   
   
   
   
6. If there is a b in R^n such that the equation Ax=b is consistent, where A is nxn, then the solution is unique.
   a) true
   b) false and I can think of a counterexample
   c) false and I can think of a correction
   d) both b) and c)
   e) other

1. e)
2. d)
3. e) (a and c--b should say columns)
4. c)
5. d)
6. d)