31. The transformation matrix for mirroring a point in x-y plane about the line y = x is given by
32. For the matrix, \[\left[ {\begin{array}{*{20}{c}}
4&1 \\
1&4
\end{array}} \right]\] the eigen values are
33. For the matrix A satisfying the equation given below, the eigen values are
\[\left[ {\text{A}} \right]\left[ {\begin{array}{*{20}{c}}
1&2&3 \\
7&8&9 \\
4&5&6
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
1&2&3 \\
4&5&6 \\
7&8&9
\end{array}} \right]\]
\[\left[ {\text{A}} \right]\left[ {\begin{array}{*{20}{c}} 1&2&3 \\ 7&8&9 \\ 4&5&6 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&2&3 \\ 4&5&6 \\ 7&8&9 \end{array}} \right]\]
34. The matrix \[{\text{M}} = \left[ {\begin{array}{*{20}{c}}
{ - 2}&2&{ - 3} \\
2&1&{ - 6} \\
{ - 1}&{ - 2}&0
\end{array}} \right]\] has eigen values -3, -3, 5. An eigen vector corresponding to the eigen value 5 is [1 2 -1]T. One of the eigen vectors of the matrix M3 is
35. The inverse of the matrix \[\left[ {\begin{array}{*{20}{c}}
{3 + 2{\text{i}}}&{\text{i}} \\
{ - {\text{i}}}&{3 - 2{\text{i}}}
\end{array}} \right]\] is
36. In matrix equation [A]{X} = {R}
\[\left[ {\text{A}} \right] = \left[ {\begin{array}{*{20}{c}}
4&8&4 \\
8&{16}&{ - 4} \\
4&{ - 4}&{15}
\end{array}} \right],\,\left\{ {\text{X}} \right\} = \left\{ {\begin{array}{*{20}{c}}
2 \\
1 \\
4
\end{array}} \right\}\,{\text{and }}\left\{ {\text{R}} \right\} = \left\{ {\begin{array}{*{20}{c}}
{32} \\
{16} \\
{64}
\end{array}} \right\}\]
One of the eigen values of Matrix [A] is
\[\left[ {\text{A}} \right] = \left[ {\begin{array}{*{20}{c}} 4&8&4 \\ 8&{16}&{ - 4} \\ 4&{ - 4}&{15} \end{array}} \right],\,\left\{ {\text{X}} \right\} = \left\{ {\begin{array}{*{20}{c}} 2 \\ 1 \\ 4 \end{array}} \right\}\,{\text{and }}\left\{ {\text{R}} \right\} = \left\{ {\begin{array}{*{20}{c}} {32} \\ {16} \\ {64} \end{array}} \right\}\]
One of the eigen values of Matrix [A] is
37. Consider a 2 × 2 square matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}}
\sigma &{\text{x}} \\
\omega &\sigma
\end{array}} \right]\] where x is unknown. If the eigen values of the matrix A are \[\left( {\sigma + {\text{j}}\omega } \right)\] and \[\left( {\sigma - {\text{j}}\omega } \right)\] , then x is equal to
38. Solution for the system defined by the set of equations 4y + 3z = 8; 2x - z = 2 and 3x + 2y = 5 is
39. The solution to the system of equations is \[\left[ {\begin{array}{*{20}{c}}
2&5 \\
{ - 4}&3
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{\text{x}} \\
{\text{y}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
2 \\
{ - 30}
\end{array}} \right]\]
40. For the given orthogonal matrix Q
\[{\text{Q}} = \left[ {\begin{array}{*{20}{c}}
{\frac{3}{7}}&{\frac{2}{7}}&{\frac{6}{7}} \\
{ - \frac{6}{7}}&{\frac{3}{7}}&{\frac{2}{7}} \\
{\frac{2}{7}}&{\frac{6}{7}}&{ - \frac{3}{7}}
\end{array}} \right]\]
The inverse is
\[{\text{Q}} = \left[ {\begin{array}{*{20}{c}} {\frac{3}{7}}&{\frac{2}{7}}&{\frac{6}{7}} \\ { - \frac{6}{7}}&{\frac{3}{7}}&{\frac{2}{7}} \\ {\frac{2}{7}}&{\frac{6}{7}}&{ - \frac{3}{7}} \end{array}} \right]\]
The inverse is
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