A set of linear equations is represented by the matrix equations $$Ax=b.$$ The necessary condition for the existence of a solution for this system is

If $$A = \left[ {\matrix{ 5 & 0 & 2 \cr 0 & 3 & 0 \cr 2 & 0 & 1 \cr } } \right]$$ then $${A^{ - 1}} = $$

Express the given matrix $$A = \left[ {\matrix{ 2 & 1 & 5 \cr 4 & 8 & {13} \cr 6 & {27} & {31} \cr } } \right]$$
as a product of triangular matrices $$L$$ and $$U$$ where the diagonal elements of the lower triangular matrices $$L$$ are unity and $$U$$ is an upper triangular matrix.

The inverse of the matrix $$S = \left[ {\matrix{ 1 & { - 1} & 0 \cr 1 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right]$$ is