Consider solving the following system of simultaneous equations using LU decomposition.

x₁ + x₂ $$-$$ 2x₃ = 4

x₁ + 3x₂ $$-$$ x₃ = 7

2x₁ + x₂ $$-$$ 5x₃ = 7

where L and U are denoted as

$$L = \left( {\matrix{ {{L_{11}}} & 0 & 0 \cr {{L_{21}}} & {{L_{22}}} & 0 \cr {{L_{31}}} & {{L_{32}}} & {{L_{33}}} \cr } } \right),\,U = \left( {\matrix{ {{U_{11}}} & {{U_{12}}} & {{U_{13}}} \cr 0 & {{U_{22}}} & {{U_{23}}} \cr 0 & 0 & {{U_{33}}} \cr } } \right)$$

Which one of the following is the correct combination of values for L₃₂, U₃₃, and x₁ ?

Which of the following is/are the eigenvector(s) for the matrix given below?

$$\left( {\matrix{ { - 9} & { - 6} & { - 2} & { - 4} \cr { - 8} & { - 6} & { - 3} & { - 1} \cr {20} & {15} & 8 & 5 \cr {32} & {21} & 7 & {12} \cr } } \right)$$

For two n-dimensional real vectors P and Q, the operation s(P, Q) is defined as follows:

$$s\left( {P,\;Q} \right) = \mathop \sum \limits_{i = 1}^n \left( {p\left[ i \right].Q\left[ i \right]} \right)$$

Let L be a set of 10-dimensional non-zero vectors such that for every pair of distinct vectors P, Q ∈ L, s(P, Q) = 0. What is the maximum cardinality possible for the set L ?

Consider the following matrix.

$$\left( {\begin{array}{*{20}{c}} 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&0 \end{array}} \right)$$

The largest eigenvalue of the above matrix is ______