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Department of Mathematics & Statistics
University of North Florida
Prof. Champak D. Panchal
March 14, 2005
MAS 3105, TEST II, NAME ________________ , _________________
1 For
2 1 3 


A  1 2 0  find the inverse matrix and using this inverse solve
 3  2 5


2 x1  x 2  3x3  10
x1  2 x 2
 20
3x1  2 x 2  5 x3  60
 1 1 2 


2.Find LU decomposition of A    2 4  1 , write elementary matrices that you use
 5  1 15 


in this work, and also write down the inverses of these elementary matrices.
3.(i) Find a, b, and c if
 2  1    1  o 
       
a1   b 2   c 0    o  Are three vectors on L.. H . S . linearly indepedent ?
 3   21  5   o 
       
1 
  1
 2  0
 
 
   
(i ) Find a, b, c if a  2   b 4   c  7    0 
 5 
  1
 5  0
 
 
   
Are three vectors on L.. H . S . linearly indepedent ?
4. (i)
 2x  3y 

x 
If T     x  2 y , justify that T ( x   y )   T x   T y, where
 y   2 y  5x 


x and y are vectors in R 2 .
(ii) Find the matrix that represents the linear transformation in (i) above.
Department of Mathematics & Statistics
University of North Florida
Prof. Champak D. Panchal
March 14, 2005
5.(i) The matrices A and Q are of size n by n. Q is invertible, then we have proved that
Rank(A) < Rank(AQ). Using this fact, prove Rank(A) = Rank(AB), where
B is an n by n invertible matrix.
(ii)For B in (i) above prove that ( B T ) 1  ( B 1 ) T


 
 u belong to Span u, v, w?
 
(iii ) Justify that Span u, v, z  Span u, v iff z is in Span u, v
(iv) (i ) Does
(v)
A is a 20 by 30 matrix, fill in the following blanks:
(a) A maps R
into R
.
(b) -------- < Rank(A) < --------(c) Rank(A) ------- Rank(A ).