$$\sum\limits_{x = 1}^{99} {{1 \over {x\left( {x + 1} \right)}}} $$ = _____________.

Consider the following $$2 \times 2$$ matrix $$A$$ where two elements are unknown and are marked by $$a$$ and $$b.$$ The eigenvalues of this matrix ar $$-1$$ and $$7.$$ What are the values of $$a$$ and $$b$$?
$$A = \left( {\matrix{ 1 & 4 \cr b & a \cr } } \right)$$

Perform the following operations on the matrix $$\left[ {\matrix{ 3 & 4 & {45} \cr 7 & 9 & {105} \cr {13} & 2 & {195} \cr } } \right]$$
(i) Add the third row to the second row
(ii) Subtract the third column from the first column.

The determinant of the resultant matrix is _________.

The product of the non-zero eigenvalues of
the matrix $$\left[ {\matrix{ 1 & 0 & 0 & 0 & 1 \cr 0 & 1 & 1 & 1 & 0 \cr 0 & 1 & 1 & 1 & 0 \cr 0 & 1 & 1 & 1 & 0 \cr 1 & 0 & 0 & 0 & 1 \cr } } \right]$$ is ____ .