Question

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

• A. $\left(\frac{3^{10}-3}{2}\right) A$
• B. $\left(\frac{3^{10}-1}{2}\right) A$
• C. $\left(\frac{3^{10}+1}{2}\right) A$
• D. $\left(\frac{3^{10}+3}{2}\right) A$

Answer 1

Answer: (A)

$A =\begin{pmatrix} 1 & 2 & 2^{2} \\ 1 / 2 & 1 & 2 \\ 1 / 2^{2} & 1 / 2 & 1\end{pmatrix}$
$A ^{2}=3 A$
$A ^{3}=3^{2} A$
$A ^{2}+ A ^{3}+\ldots . A ^{10}$
$=3 A +3^{2} A +\ldots+3^{9} A =\frac{3\left(3^{9}-1\right)}{3-1} A$
$=\frac{3^{10}-3}{2} A$