Pretest 1.

This "pretest" is similar to the first test, Feb. 12. Remember that you need to justify each step in your solution. You can use Maple and can send me the solutions by e-mail. The WebNotes will be available but some links to proofs may be temporarily unavailable. In order to prepare for the test you need to look at the proofs of all theorems and lemmas that we already proved and at the solutions of all homeworks available on the Web. Your opinions about this pretest are welcome on the discussion page.

Solutions of the pretest 1
Maple HTML.

***To make the matrices in the Maple worksheet look better, make the window with the worksheet wider.
  1. Prove the last statement in the theorem about transposes ((AB)T=BTAT).
  2. Find the general solution of the following system of linear equations depending on a parameter a: 
    
				         ax+y+z = 2
				          x-y+z = 3
				        5x+y+3z = 7
  3. Find the determinant of the following n by n matrix
    [ 1 2 0 ... 0 0 ]
    [ -1 2 0 ... 0 0 ]
    [ 0 -1 3 ... 0 0 ]
    ........................
    [ 0 0 0 ... n-1 0 ]
    [ 0 0 0 ... -1 n ]
    (cofactor expansion along the first row).
  4. Solve the following system of equations using the Kramer's rule: 
    
			           4x + 5y + 6z = 10
			           6x - 2y + 7z = 20
			           7x + 5y - 2z = 9