Page 10 - Engineering Mathematics Workbook_Final

Linear Algebra

(a)  a a a 
  1 2 3 


   x 0 (x x− 1 )(x x− 2 )(x x− 3 )(x x− 4  )   4 42. Let A be the matrix b 1 b 2 b  c 3 3   

c
c


1
)
−
−
−
−
x
(b) ( x x 1 )( x x 2 )( x x 3 )( x x where a i a j a k , 2 
+
+
0
4
3
1
2
+
+
+
+
(c) b i b j b k , c i c j c k are
2
1
2
3
3
1
−
−
−
−
x 0    (x x 1 )(x x 2 )(x x 3 )(x x 4  )   4 three mutually orthogonal unit vector
matrix . Then ________
(d) x x x x x x
4
3
2
1
0
−
2
1
(a) A = A (b) A = A
[IISC 2005]

−1

(c) = (d) =
40. The determinant [IISC 2004]
+
1 x 1 1 1
−
1 1 x 1 1 43. For real numbers a, b, c the following
linear system of equations
+
1 1 1 y 1 x + + = 1, ax by cz = 1,
+
+
y z
−
1 1 1 1 y
a x b y c z = 1 has a unique
2
+
2
+
2
(a) xy solution if and only if _________

=
(a) b a and b c
2
xy
(b) ( )

=
(b) a b and a c
−
−
(c) (1 x 2 )(1 y 2 ) (c) a c and a b
=




2
2
(d) x + y [IISC 2005] (d) a b b c and a c
41. Let a, b, c are arbitrary real numbers. Let [IISC 2003]
 1 a b 


  44. Let P be a 3 2 matrix, Q be a 2 2

A be the matrix A =   0 1 c  . Let I matrix and R be a 2 3 matrix such
 
   0 0 1    that PQR is equal to the identity matrix.

be the 3 3 identity matrix. Then Then ____
(a) rank of P is equal to 2
1
2
(a) A − 3A + 3I = A
(b) Q is non-singular
1
2
(b) A + 3A + 3I = A (c) Both A and B are true
1
+
2
(c) A + A I = A (d) There are no such matrices P, Q and

[IISC 2012]

R
(d) A is not invertible [IISC 2004]

8