41. The two vectors [1 1 1] and [1, a, a2], where \[{\text{a}} = \left( { - \frac{1}{2} + {\text{j}}\frac{{\sqrt 3 }}{2}} \right)\] , are
42. Multiplication of matrices E and F is G. Matrices E and G are \[{\text{E}} \equiv \left[ {\begin{array}{*{20}{c}}
{\cos \theta }&{ - \sin \theta }&0 \\
{\sin \theta }&{\cos \theta }&0 \\
0&0&1
\end{array}} \right]{\text{and G}} \equiv \left[ {\begin{array}{*{20}{c}}
1&0&0 \\
0&1&0 \\
0&0&1
\end{array}} \right].\]
What is the matrix F?
What is the matrix F?
43. If a matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}}
2&4 \\
1&3
\end{array}} \right]\] and matrix \[{\text{B}} = \left[ {\begin{array}{*{20}{c}}
4&6 \\
5&9
\end{array}} \right]\] the transpose of product of these two matrices i.e., (AB)T is
44. Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigen values of this matrix are -1 and 7. What are the values of a and b?
\[{\text{A}} = \left( {\begin{array}{*{20}{c}}
1&4 \\
{\text{b}}&{\text{a}}
\end{array}} \right)\]
\[{\text{A}} = \left( {\begin{array}{*{20}{c}} 1&4 \\ {\text{b}}&{\text{a}} \end{array}} \right)\]
45. 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
2x1 - x2 + 3x3 = 1
3x1 - 2x2 + 5x3 = 2
-x1 - 4x2 + x3 = 3
This system of equations has
46. Consider the matrix
\[{\text{P}} = \left[ {\begin{array}{*{20}{c}}
1&1&0 \\
0&1&1 \\
0&0&1
\end{array}} \right]\]
The number of distinct eigen values of P is
\[{\text{P}} = \left[ {\begin{array}{*{20}{c}} 1&1&0 \\ 0&1&1 \\ 0&0&1 \end{array}} \right]\]
The number of distinct eigen values of P is
47. The system of linear equations
4x + 2y = 7
2x + y = 6
has
4x + 2y = 7
2x + y = 6
has
48. For what value of a, if any, will the following system of equations in x, y and z have a solution?
2x + 3y = 4; x + y + z = 4; x + 2y - z = a
2x + 3y = 4; x + y + z = 4; x + 2y - z = a
49. The eigen value of the following matrix \[\left[ {\begin{array}{*{20}{c}}
{10}&{ - 4} \\
{18}&{ - 12}
\end{array}} \right]\]
50. Given an orthogonal matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}}
1&1&1&1 \\
1&1&{ - 1}&{ - 1} \\
1&{ - 1}&0&0 \\
0&0&1&{ - 1}
\end{array}} \right],\,{\left[ {{\text{A}}{{\text{A}}^{\text{T}}}} \right]^{ - 1}}\,{\text{is}}\]
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