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  4. If A = [0&5 0&0] and f(X) =1+x+x2 +....+x16 , then f(A) =

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Q. If $A = \begin{bmatrix}0&5\\ 0&0\end{bmatrix} $ and $f\left(X\right) =1+x+x^{2} +....+x^{16} $, then $f(A) = $

Matrices

Solution:

Clearly $f(A)=I+A+A^2+......+A^{16}$
$A^{2} = AA = \begin{bmatrix}0&5\\ 0&0\end{bmatrix} \begin{bmatrix}0&5\\ 0&0\end{bmatrix} = \begin{bmatrix}0&0\\ 0&0\end{bmatrix}= O $
$A^3=0,A^4=0,.......A^{16}=0$
$\therefore $$f(A)$$ = \begin{bmatrix}1&0\\ 0&1\end{bmatrix} + \begin{bmatrix}0&5\\ 0&0\end{bmatrix}+0+0+.....+0$
$=\begin{bmatrix}1&5\\ 0&1\end{bmatrix}$