For a matrix \[\left[ {\text{M}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{3}{5}}&{\frac{4}{5}} \\ {\text{x}}&{\frac{3}{5}} \end{array}} \right],\]    the transpose of the matrix is equal to the inverse of the matrix, [M]^T = [M]^-1. The value of x is given by

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?
\[\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&0 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 0&1 \\ 0&0 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ 1&1 \end{array}} \right]{\text{ and }}\left[ {\begin{array}{*{20}{c}} { - 1}&0 \\ 1&{ - 1} \end{array}} \right]\]