The lowest eigen value of the $$2 \times 2$$ matrix $$\left[ {\matrix{ 4 & 2 \cr 1 & 3 \cr } } \right]$$ is ______.

Given that the determinant of the matrix $$\left[ {\matrix{ 1 & 3 & 0 \cr 2 & 6 & 4 \cr { - 1} & 0 & 2 \cr } } \right]$$ is $$-12$$, the determinant of the matrix $$\left[ {\matrix{ 2 & 6 & 0 \cr 4 & {12} & 8 \cr { - 2} & 0 & 4 \cr } } \right]$$ is

Which one of the following describes the relationship among the three vectors, $$\widehat i + \widehat j + \widehat k,\,\,2\widehat i + 3\widehat j + \widehat k$$ and $$5\widehat i + 6\widehat j + 4\widehat k?$$

Which one of the following equations is a correct identity for arbitrary $$3 \times 3$$ real matrices $$P,Q$$ and $$R$$?