$$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right] + \left[ {\matrix{ 0 & a & a \cr 0 & 0 & b \cr 0 & 0 & 0 \cr } } \right] = I + B$$

$${B^2} = \left[ {\matrix{ 0 & a & a \cr 0 & 0 & b \cr 0 & 0 & 0 \cr } } \right] + \left[ {\matrix{ 0 & a & a \cr 0 & 0 & b \cr 0 & 0 & 0 \cr } } \right] = \left[ {\matrix{ 0 & 0 & {ab} \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right]$$

$${B^3} = 0$$

$$\therefore$$ $${A^n} = {(1 + B)^n} = {}^n{C_0}I + {}^n{C_1}B + {}^n{C_2}{B^2} + {}^n{C_3}{B^3} + \,\,....$$

$$ = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right] + \left[ {\matrix{ 0 & {na} & {na} \cr 0 & 0 & {nb} \cr 0 & 0 & 0 \cr } } \right] + \left[ {\matrix{ 0 & 0 & {{{n(n - 1)ab} \over 2}} \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right]$$

$$ = \left[ {\matrix{ 1 & {na} & {na + {{n(n - 1)} \over 2}ab} \cr 0 & 1 & {nb} \cr 0 & 0 & 1 \cr } } \right] = \left[ {\matrix{ 1 & {48} & {2160} \cr 0 & 1 & {48} \cr 0 & 0 & 1 \cr } } \right]$$

On comparing we get $$na = 48$$, $$nb = 96$$ and

$$na + {{n(n - 1)} \over 2}ab = 2160$$

$$ \Rightarrow a = 4,n = 12$$ and $$b = 8$$

$$n + a + b = 24$$