81. The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}}
4&{ - 2} \\
{ - 2}&1
\end{array}} \right]\]
82. For the set of equations
x1 + 2x2 + x3 + 4x4 = 2
3x1 + 6x2 + 3x3 + 12x4 = 6
the following statement is true:
x1 + 2x2 + x3 + 4x4 = 2
3x1 + 6x2 + 3x3 + 12x4 = 6
the following statement is true:
83. Let A be an m × n matrix and Ban n × m matrix. It is given that determinant ($$I$$m + AB) = determinant ($$I$$n + BA), where $$I$$k is the k × k identity matrix. Using the above property, the determinant of the matrix given below is
\[\left[ {\begin{array}{*{20}{c}}
2&1&1&1 \\
1&2&1&1 \\
1&1&2&1 \\
1&1&1&2
\end{array}} \right]\]
\[\left[ {\begin{array}{*{20}{c}} 2&1&1&1 \\ 1&2&1&1 \\ 1&1&2&1 \\ 1&1&1&2 \end{array}} \right]\]
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