131.  Consider the following matrix A. If the eigenvalues of A are 4 and 8, then 
a.  x=4, y=10 
b.  x=5, y=8 
c.  x=3, y=9 
d.  x=4, y=10 
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Answer: (d).x=4, y=10

132.  How many of the following matrices have an eigenvalue 1? 
a.  one 
b.  two 
c.  three 
d.  four 
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Answer: (a).one

133.  If the matrix A is such that, then the determinant of A is equal to 
a.  0 
b.  1 
c.  2 
d.  3 
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Answer: (a).0

134.  The product of the nonzero eigenvalues of the matrix 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 1 0 0 0 1 is ______ 
a.  4 
b.  5 
c.  6 
d.  7 
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Answer: (c).6

135.  Which one of the following statements is TRUE about every n×n matrix with only real eigenvalues? 
a.  If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative. 
b.  If the trace of the matrix is positive, all its eigenvalues are positive. 
c.  If the determinant of the matrix is positive, all its eigenvalues are positive. 
d.  If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive. 
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Answer: (a).If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.

136.  Consider the set H of all 3 × 3 matrices of the following type, where a, b, c, d, e and f are real numbers and abc ≠ 0. Under the matrix multiplication operation, the set H is 
a.  a group 
b.  a monoid but not a group 
c.  a semigroup but not a monoid 
d.  neither a group nor a semigroup 
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Answer: (a).a group

137.  Consider the following system of equations in three real variables x1, x2 and x3: 2x1  x2 + 3x3 = 1 3x1 2x2 + 5x3 = 2 x1 + 4x2 + x3 = 3 This system of equations has 
a.  no solution 
b.  a unique solution 
c.  more than one but a finite number of solutions 
d.  an infinite number of solutions 
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Answer: (b).a unique solution

138.  What are the eigenvalues of the following 2 × 2 matrix? 
a.  1 and 1 
b.  1 and 6 
c.  2 and 5 
d.  4 and 1 
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Answer: (b).1 and 6

139.  Let A, B, C, D be n × n matrices, each with nonzero determinant. If ABCD = 1, then B^1 is 
a.  D^1C^1A^1 
b.  CDA 
c.  ADC 
d.  Does not necessarily exist 
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Answer: (b).CDA

140.  In the LU decomposition of the matrix,  2 2   4 9  if the diagonal elements of U are both 1, then the lower diagonal entry l22 of L is 
a.  4 
b.  5 
c.  6 
d.  7 
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Answer: (b).5
