Let c₁ ..... c_n be scalars, not all zero, such that \[\sum\limits_{{\text{i}} = 1}^{\text{n}} {{{\text{c}}_{\text{i}}}{{\text{a}}_{\text{i}}}} = 0\]   where a_i are column vectors in Rⁿ. Consider the set of linear equations Ax = b where A = [a₁ ..... a_n] and \[{\text{b}} = \sum\limits_{{\text{i}} = 1}^{\text{n}} {{{\text{a}}_{\text{i}}}} .\]   The set of equations has

If $${\left[ {1,\,0,\, - 1} \right]^{\text{T}}}$$  is an eigen vector of the following matrix \[\left[ {\begin{array}{*{20}{c}} 1&{ - 1}&0 \\ { - 1}&2&{ - 1} \\ 0&{ - 1}&1 \end{array}} \right]\]   then corresponding eigen value is

The eigen values of the following matrix are \[\left[ {\begin{array}{*{20}{c}} { - 1}&3&5 \\ { - 3}&{ - 1}&6 \\ 0&0&3 \end{array}} \right]\]

Let A be the 2 × 2 matrix with elements a₁₁ = a₁₂ = a₂₁ = +1 and a₂₂ = -1. Then the eigen values of the matrix A¹⁹ are

How many of the following matrices have an eigen value 1?
\[\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&0 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 0&1 \\ 0&0 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ 1&1 \end{array}} \right]{\text{ and }}\left[ {\begin{array}{*{20}{c}} { - 1}&0 \\ 1&{ - 1} \end{array}} \right]\]