81. How many solutions does the following system of linear equations have?
-x + 5y = -1; x - y = 2; x + 3y = 3
-x + 5y = -1; x - y = 2; x + 3y = 3
82. Consider the following matrix.
\[{\text{A}} = \left[ {\begin{array}{*{20}{c}}
2&3 \\
{\text{x}}&{\text{y}}
\end{array}} \right]\]
If the eigen values of A are 4 and 8, then
\[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&3 \\ {\text{x}}&{\text{y}} \end{array}} \right]\]
If the eigen values of A are 4 and 8, then
83. Let N be a 3 by 3 matrix with real number entries. The matrix N is such that N2 = 0. The eigen values of N are
84. The eigen values of a (2 × 2) matrix X are -2 and -3. The eigen values of the matrix (X + $$I$$) (X + 5$$I$$) are
85. If the following system has non-trivial solution,
px + qy + rz = 0
qx + ry + pz = 0
rx + py + qz = 0
then which one of the following options is TRUE?
px + qy + rz = 0
qx + ry + pz = 0
rx + py + qz = 0
then which one of the following options is TRUE?
86. The sum of the eigen values of the matrix given below is \[\left[ {\begin{array}{*{20}{c}}
1&2&3 \\
1&5&1 \\
3&1&1
\end{array}} \right].\]
87. The following system of equations
x1 + x2 + 2x3 = 1
x1 + 2x3 + 3x3 = 2
x1 + 4x2 + ax3 = 4
has a unique solution. The only possible value(s) for a is/are
x1 + x2 + 2x3 = 1
x1 + 2x3 + 3x3 = 2
x1 + 4x2 + ax3 = 4
has a unique solution. The only possible value(s) for a is/are
88. The eigen vector pair of the matrix \[\left[ {\begin{array}{*{20}{c}}
3&4 \\
4&{ - 3}
\end{array}} \right]\] is
89. For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold?
90. The following vector is linearly dependent upon the solution to the previous problem
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