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Mathematics Question on Matrices

If A=[05 00]A = \begin{bmatrix}0&5\\\ 0&0\end{bmatrix} and f(X)=1+x+x2+....+x16f\left(X\right) =1+x+x^{2} +....+x^{16} , then f(A)=f(A) =

A

OO

B

[15 01]\begin{bmatrix} 1 &5\\\ 0& 1 \end{bmatrix}

C

[15 00]\begin{bmatrix} 1 &5\\\ 0&0\end{bmatrix}

D

[05 11]\begin{bmatrix}0&5\\\ 1 &1\end{bmatrix}

Answer

[15 01]\begin{bmatrix} 1 &5\\\ 0& 1 \end{bmatrix}

Explanation

Solution

Clearly f(A)=I+A+A2+......+A16f(A)=I+A+A^2+......+A^{16} A2=AA=[05 00][05 00]=[00 00]=OA^{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 A3=0,A4=0,.......A16=0A^3=0,A^4=0,.......A^{16}=0 ∴\therefore f(A)f(A) =[10 01]+[05 00]+0+0+.....+0 = \begin{bmatrix}1&0\\\ 0&1\end{bmatrix} + \begin{bmatrix}0&5\\\ 0&0\end{bmatrix}+0+0+.....+0 =[15 01]=\begin{bmatrix}1&5\\\ 0&1\end{bmatrix}