Let A=[a_ij] be a square matrix of order 3 such that a_ij=2^j-i for all i,j=1,2,3.

Question

Let $A=[a_{ij}]$ be a square matrix of order 3 such that $a_{ij}=2^{j-i}$ for all $i,j=1,2,3$. Then the matrix $A^2+A^3+\cdots+A^{10}$ is equal to:

Accepted Answer

$A=\begin{pmatrix}1&2&4\\frac12&1&2\\frac14&\frac12&1\end{pmatrix}$. Direct computation shows $A^2=3A$, and by induction $A^k=3^{k-1}A$.

$$A^2+A^3+\cdots+A^{10}=A(3+3^2+\cdots+3^9)=A\cdot\frac{3(3^9-1)}{2}=\frac{3^{10}-3}{2}A.$$

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