1. Let A be an n × n real matrix such that A2 = $$I$$ 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
2. Let A = [aij],1 ≤ i, j ≤ n with n ≥ 3 and aij = i.j. The rank of A is
3. All 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?
4. The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}}
0&1 \\
{ - 1}&0
\end{array}} \right]\] are
5. If the rank of a (5 × 6) matrix Q is 4, then which one of the following statements is correct?
6. Consider 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
7. Let A be an n × n matrix with rank r(0 < r < n). Then AX = 0 has p independent solutions, where p is
8. The 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
9. In 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
10. If 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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