21. The eigen values of a symmetric matrix are all
22. Which one of the following equations is a correct identity for arbitrary 3 × 3 real matrices P, Q and R?
23. The rank of the matrix \[\left[ {\begin{array}{*{20}{c}}
1&1&1 \\
1&{ - 1}&0 \\
1&1&1
\end{array}} \right]\] is
24. It is given that X1, X2, ... XM are M non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M vectors X1, X2 ... XM, -X1, -X2 ... -XM is
25. Consider the set of (column) vectors defined by X = {x \[ \in \] R3 | x1 + x2 + x3 = 0, where xT =[x1, x2, x3]T}. Which of the following is TRUE?
26. For given matrix \[{\text{P}} = \left[ {\begin{array}{*{20}{c}}
{4 + 3{\text{i}}}&{ - {\text{i}}} \\
{\text{i}}&{4 - 3{\text{i}}}
\end{array}} \right]\] where \[{\text{i}} = \sqrt { - 1} ,\] the inverse of matrix P is
27. The matrix \[\left[ {\begin{array}{*{20}{c}}
1&2&4 \\
3&0&6 \\
1&1&{\text{p}}
\end{array}} \right]\] has one eigen value equal to 3. The sum of the other two eigen values is
28. The linear operation L(x) is defined by the cross product L(x) = b × X, where b = [0 1 0]T and X = [x1x2x3]T are three dimensional vectors. The 3 × 3 matrix M of this operation satisfies \[{\text{L}}\left( {\text{x}} \right) = {\text{M}}\left[ {\begin{array}{*{20}{c}}
{{{\text{x}}_1}} \\
{{{\text{x}}_2}} \\
{{{\text{x}}_3}}
\end{array}} \right].\]
Then the eigen values of M are
Then the eigen values of M are
29. What are the value of k for which the system of equations:
(3k - 8)x + 3y + 3z = 0
3x + (3k - 8)y + 3z = 0
3x + 3y + (3k - 8)z = 0
has a not-trivial solution?
(3k - 8)x + 3y + 3z = 0
3x + (3k - 8)y + 3z = 0
3x + 3y + (3k - 8)z = 0
has a not-trivial solution?
30. The rank of the following matrix is \[\left( {\begin{array}{*{20}{c}}
1&1&0&{ - 2} \\
2&0&2&2 \\
4&1&3&1
\end{array}} \right)\]
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