The matrix $$\left[ A \right] = \left[ {\matrix{ 2 & 1 \cr 4 & { - 1} \cr } } \right]$$ is decomposed into a product of lower triangular matrix $$\left[ L \right]$$ and an upper triangular $$\left[ U \right].$$ The properly decomposed $$\left[ L \right]$$ and $$\left[ U \right]$$ matrices respectively are

Roots of the algebraic equation $${x^3} + {x^2} + x + 1 = 0$$ are

The function $$f\left( x \right) = 2x - {x^2} + 3\,\,$$ has

The two vectors $$\left[ {\matrix{ {1,} & {1,} & {1} \cr } } \right]$$ and $$\left[ {\matrix{ {1,} & {a,} & {{a^2}} \cr } } \right]$$ where $$a = {{ - 1} \over 2} + j{{\sqrt 3 } \over 2}$$ are