The rank of the following (n + 1) x (n + 1) matrix, where a is a real number is $$$\left[ {\matrix{ 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr . & . & . & . & . & . & . \cr . & . & . & . & . & . & . \cr . & . & . & . & . & . & . \cr 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr } } \right]$$$

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

The rank of the matrix $$\left[ {\matrix{ 0 & 0 & { - 3} \cr 9 & 3 & 5 \cr 3 & 1 & 1 \cr } } \right]$$ is

If $$A = \left[ {\matrix{ 1 & 0 & 0 & 1 \cr 0 & { - 1} & 0 & { - 1} \cr 0 & 0 & i & i \cr 0 & 0 & 0 & { - i} \cr } } \right]$$ the matrix $${A^4},$$
calculated by the use of Cayley - Hamilton theoram (or) otherwise is