A sequence $$x\left[ n \right]$$ is specified as $$$\left[ {\matrix{ {x\left[ n \right]} \cr {x\left[ {n - 1} \right]} \cr } } \right] = {\left[ {\matrix{ 1 & 1 \cr 1 & 0 \cr } } \right]^n}\left[ {\matrix{ 1 \cr 0 \cr } } \right],\,\,for\,\,n \ge 2.$$$
The initial conditions are $$x\left[ 0 \right] = 1,\,\,x\left[ 1 \right] = 1$$ and $$x\left[ n \right] = 0$$ for $$n < 0.$$ The value of $$x\left[ {12} \right]$$ is __________.

If the vectors $${e_1} = \left( {1,0,2} \right),\,{e_2} = \left( {0,1,0} \right)$$ and $${e_3} = \left( { - 2,0,1} \right)$$ form an orthogonal basis of the three dimensional real space $${R^3},$$ then the vectors $$u = \left( {4,3, - 3} \right) \in {R^3}$$ can be expressed as

The eigen values of the following matrix $$\left[ {\matrix{ { - 1} & 3 & 5 \cr { - 3} & { - 1} & 6 \cr 0 & 0 & 3 \cr } } \right]$$ are

The eigen values and the correspondinng eigen vectors of a $$2 \times 2$$ matrix are given by

Eigen value
$${\lambda _1} = 8$$
$${\lambda _2} = 4$$

Eigen vector
$${V_1} = \left[ {\matrix{ 1 \cr 1 \cr } } \right]$$
$${V_2} = \left[ {\matrix{ 1 \cr -1 \cr } } \right]$$

The matrix is