Page 4 - Engineering Mathematics Workbook_Final

Linear Algebra


1. Let A be the n n matrix with   12 24 5  
−
1
integer entries and assume that A also 4. Let A =   x 6 2  . The value of
has only integer entries, then     − 1 − 2 3    

(a) det A = n x for which the matrix A is not
invertible is ________
(b) det A =  1
(a) 6 (b) 12
=
2
(c) det A n (c) 3 (d) 2
(d) det A will depend on the entries of A [IISC 2007, IISC 2008]
−
1
and A
  p 
[IISC 2007] 5. Let A =     where p, q, r are
  q r  

2. Let A be a 3 3 matrix with Eigen rational numbers. A = the value of
0
−
values 1, 1, and 3. Then __________
2
2
+ ______
2
(a) A + A is non-singular
(a) 2
2
(b) A − A is non-singular (b) 1
2
(c) A + 3A is non-singular (c) 0

2
(d) A − 3A is non-singular (d) cannot be determine using given data
f
[JAM 2005] 6. For n > 1, let ( ) n be the number of

 cos  − sin   n n real matrices A such that
3. The Eigen values of     A + 2 I = 0. Then
 sin   cos   
are (a) f = 0

(a) cos and sin (b) ( ) 0f n = its n is even

(b) tan and cot
(c) ( ) 0f n = its n is odd
i
(c) e and e − i
(d) f = 
(d) 1 and 2
[IISC 2007]
[IISC 2007]
 a b 
7. Let A =     be a 2 2 real
  c d 

matrix with det A = 1. If A has no real
Eigen values then

2