61
X = [x1 , x2 , ... xn ]T is an n-tuple nonzero vector. The n × n matrix V = XXT
A)
has rank zero
B)
has rank 1
C)
is orthogonal
D)
has rank n
Answer & Solution
Answer: Option
B
62
Which one of the following statements is NOT true for a square matrix A?
A)
If A is upper triangular, the eigen values of A are the diagonal elements of it
B)
If A is real symmetric, the eigen values of A are always real and positive
C)
If A is real, the eigen values of A and AT are always the same
D)
If all the principal minors of A are positive, all the eigen values of A are also positive
Answer & Solution
Answer: Option
B
63
For any real, square and non-singular matrix B, the detB-1 is
A)
zero
B)
(detB)-1
C)
- (detB)
D)
detB
Answer & Solution
Answer: Option
B
64
Given a system of equations:1 2
A)
The system has a unique solution for any given b1 and b2
B)
The system will have infinitely many solutions for any given b1 and b2
C)
Whether or not a solution exists depends on the given b1 and b2
D)
The system would have no solution for any values of b1 and b2
Answer & Solution
Answer: Option
B
65
The matrix \[\left( {\begin{array}{*{20}{c}}
2&{ - 4} \\
4&{ - 2}
\end{array}} \right)\] has
A)
real eigenvalues and eigenvectors
B)
real eigenvalues but complex eigenvectors
C)
complex eigenvalues but real eigenvectors
D)
complex eigenvalues and eigenvectors
Answer & Solution
Answer: Option
D
66
The inverse of the matrix \[\left[ {\begin{array}{*{20}{c}}
2&3&4 \\
4&3&1 \\
1&2&4
\end{array}} \right]\] is
A)
\[\left[ {\begin{array}{*{20}{c}}
{10}&{ - 4}&{ - 9} \\
{ - 15}&4&{14} \\
5&{ - 1}&{ - 6}
\end{array}} \right]\]
B)
\[\left[ {\begin{array}{*{20}{c}}
{ - 10}&4&9 \\
{15}&{ - 4}&{ - 14} \\
{ - 5}&1&6
\end{array}} \right]\]
C)
\[\left[ {\begin{array}{*{20}{c}}
2&{ - \frac{4}{5}}&{ - \frac{9}{5}} \\
{ - 3}&{\frac{4}{5}}&{\frac{{14}}{5}} \\
1&{ - \frac{1}{5}}&{ - \frac{6}{5}}
\end{array}} \right]\]
D)
\[\left[ {\begin{array}{*{20}{c}}
{ - 2}&{\frac{4}{5}}&{\frac{9}{5}} \\
3&{ - \frac{4}{5}}&{ - \frac{{14}}{5}} \\
{ - 1}&{\frac{1}{5}}&{\frac{6}{5}}
\end{array}} \right]\]
Answer & Solution
Answer: Option
D
67
The condition for which the eigen values of the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}}
2&1 \\
1&{\text{k}}
\end{array}} \right]\] are positive, is
A)
k > \[\frac{1}{2}\]
B)
k > -2
C)
k > 0
D)
k < -\[\frac{1}{2}\]
Answer & Solution
Answer: Option
A
68
Given that the determinant of the matrix \[\left[ {\begin{array}{*{20}{c}}
1&3&0 \\
2&6&4 \\
{ - 1}&0&2
\end{array}} \right]\] is -12, the determinant of the matrix \[\left[ {\begin{array}{*{20}{c}}
2&6&0 \\
4&{12}&8 \\
{ - 2}&0&4
\end{array}} \right]\] is
A)
-96
B)
-24
C)
24
D)
96
Answer & Solution
Answer: Option
A
69
For the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}}
5&3 \\
1&3
\end{array}} \right],\] ONE of the normalized eigen vectors is given as
A)
\[\left( {\begin{array}{*{20}{c}}
{\frac{1}{2}} \\
{\frac{{\sqrt 3 }}{2}}
\end{array}} \right)\]
B)
\[\left( {\begin{array}{*{20}{c}}
{\frac{1}{{\sqrt 2 }}} \\
{\frac{{ - 1}}{{\sqrt 2 }}}
\end{array}} \right)\]
C)
\[\left( {\begin{array}{*{20}{c}}
{\frac{3}{{\sqrt {10} }}} \\
{\frac{{ - 1}}{{\sqrt {10} }}}
\end{array}} \right)\]
D)
\[\left( {\begin{array}{*{20}{c}}
{\frac{1}{{\sqrt 5 }}} \\
{\frac{2}{{\sqrt 5 }}}
\end{array}} \right)\]
Answer & Solution
Answer: Option
B
70
The product of eigen values of the matrix P is \[{\text{P}} = \left[ {\begin{array}{*{20}{c}}
2&0&1 \\
4&{ - 3}&3 \\
0&2&{ - 1}
\end{array}} \right]\]
A)
-6
B)
2
C)
6
D)
-2
Answer & Solution
Answer: Option
B
