Let A=left[a_{i j}right] be a square matrix of order 3 such that a_{i j}=2^{j-i} , for all i, j=1,2,

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3 and 4 .Determinants and Matrices

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Let $A=\left[a_{i j}\right]$ 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

$\left(\frac{3^{10}+3}{2}\right) A$

(JEE MAIN-2022)

Solution

$A=\left(\begin{array}{lll}1 & 2 & 2^{2} \\ 1 / 2 & 1 & 2 \\ 1 / 2^{2} & 1 / 2 & 1\end{array}\right)$

$A^{2}=3 A$

$A ^{3}=3^{2} A$

$A ^{2}+ A ^{3}+\ldots A ^{10}$

$=3 A +3^{2} A +\ldots+3^{9} A =\frac{3\left(3^{9}-1\right)}{3-1} A$

$=\frac{3^{10}-3}{2} A$

Standard 12

Mathematics

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