JEE Main 2025 · Mathematics · Matrices and Determinants · medium

Let A=beginbmatrix1&2\\0&1endbmatrix and P=A+A^2+…+A^10. The sum of all entries of P is

Single correct option

Let

A=\begin{bmatrix}1&2\\0&1\end{bmatrix}

and

P=A+A^2+\cdots+A^{10}

. The sum of all entries of

P

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Answer: (D) $130$

By induction

A^k=\begin{bmatrix}1&2k\\0&1\end{bmatrix}

. Adding for

k=1,\dots,10

gives

P=\begin{bmatrix}10&2\cdot55\\0&10\end{bmatrix}=\begin{bmatrix}10&110\\0&10\end{bmatrix}

because

\sum k=55

. The sum of entries is

10+110+0+10=130

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