Q101
Open questionLet A be an n × n real matrix such that A2 = and y be an n-dimensional vector.
Then the linear system of equations Ax = y has
Then the linear system of equations Ax = y has
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Q102
Open questionLet A = [aij],1 ≤ i, j ≤ n with n ≥ 3 and aij = i.j. The rank of A is
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Q103
Open questionAll the four entries of the 2 × 2 matrix {\text{P}} = \left[ {\begin{array}{*{20}{c}}
{{{\text{p}}_{11}}}&{{{\text{p}}_{12}}} \\
{{{\text{p}}_{21}}}&{{{\text{p}}_{22}}}
\end{array}} \right] are nonzero, and one of its eigen values is zero. Which of the following statements is true?
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Q104
Open questionThe eigen values of the matrix \left[ {\begin{array}{*{20}{c}}
0&1 \\
{ - 1}&0
\end{array}} \right] are
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Q105
Open questionIf the rank of a (5 × 6) matrix Q is 4, then which one of the following statements is correct?
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Q106
Open questionConsider the following system of equations
2x1 + x2 + x3 = 0
x2 - x3 = 0
x1 + x2 = 0
This system has
2x1 + x2 + x3 = 0
x2 - x3 = 0
x1 + x2 = 0
This system has
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Q107
Open questionLet A be an n × n matrix with rank r(0 < r < n). Then AX = 0 has p independent solutions, where p is
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Q108
Open questionThe maximum value of "a" such that the matrix \left( {\begin{array}{*{20}{c}}
{ - 3}&0&{ - 2} \\
1&{ - 1}&0 \\
0&{\text{a}}&{ - 2}
\end{array}} \right) has three linearly independent real eigen vectors is
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Q109
Open questionIn the matrix equation Px = q, which of the following is a necessary condition for the existence of at least one solution for the unknown vector x
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Q110
Open questionIf the system
2x - y + 3z = 2
x + y + 2z = 2
5x - y + az = b
has infinitely many solutions, then the values of a and b, respectively, are
2x - y + 3z = 2
x + y + 2z = 2
5x - y + az = b
has infinitely many solutions, then the values of a and b, respectively, are
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Q111
Open questionIf the vectors e1 = (1, 0, 2), e2 = (0, 1, 0) and e3 = (-2, 0, 1) form an orthogonal basis of the three-dimensional real space R3, then the vector u = (4, 3, -3) R3 can be expressed as
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Q112
Open questionA real n × n matrix A = {aij} is defined as follows: aij = i, if i = j, otherwise 0
The summation of all n eigen values of A is
The summation of all n eigen values of A is
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Q113
Open questionThe value of q for which the following set of linear equation 2x + 3y = 0; 6x + qy = 0 can have non-trivial solution is
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Q114
Open questionFor a matrix \left[ {\text{M}} \right] = \left[ {\begin{array}{*{20}{c}}
{\frac{3}{5}}&{\frac{4}{5}} \\
{\text{x}}&{\frac{3}{5}}
\end{array}} \right], the transpose of the matrix is equal to the inverse of the matrix, [M]T = [M]-1. The value of x is given by
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Q115
Open questionConsider a matrix P whose only eigenvectors are the multiples of \left[ {\begin{array}{*{20}{c}}
1 \\
4
\end{array}} \right].
Consider the following statements:
I. P does not have an inverse.
II. P has a repeated eigen value.
III. P cannot be diagonalized.
Which one of the following options is correct?
Consider the following statements:
I. P does not have an inverse.
II. P has a repeated eigen value.
III. P cannot be diagonalized.
Which one of the following options is correct?
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Q116
Open questionLet {\text{P}} = \left[ {\begin{array}{*{20}{c}}
3&1 \\
1&3
\end{array}} \right]. Consider the set S of all vectors \left( {\begin{array}{*{20}{c}}
{\text{x}} \\
{\text{y}}
\end{array}} \right) such that a2 + b2 = 1 where \left( {\begin{array}{*{20}{c}}
{\text{a}} \\
{\text{b}}
\end{array}} \right) = {\text{P}}\left( {\begin{array}{*{20}{c}}
{\text{x}} \\
{\text{y}}
\end{array}} \right). Then S is
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Q117
Open questionIn the given matrix \left[ {\begin{array}{*{20}{c}}
1&{ - 1}&2 \\
0&1&0 \\
1&2&1
\end{array}} \right], one of the eigen values is 1. The eigen vectors corresponding to the eigen value 1 are
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Q118
Open questionLet X be a square matrix. Consider the following two statements on X.
I. X is invertible.
II. Determinant of X is non-zero.
Which one of the following is TRUE?
I. X is invertible.
II. Determinant of X is non-zero.
Which one of the following is TRUE?
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Q119
Open questionChoose the CORRECT set of functions, which are linearly dependent.
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Q120
Open questionLet M4 = , (where denotes the identity matrix) and M ≠ , M2 ≠ and M3 ≠ . Then, for any natural number k, M-1 equals:
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