JEE Main 2016 · Mathematics · Matrices and Determinants · medium

Let A=beginbmatrix1&1\\0&1endbmatrix and B=A+A^2+A^3+…+A^20. The sum of all four entries of B is

Single correct option

Let

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

and

B=A+A^2+A^3+\dots+A^{20}

. The sum of all four entries of

B

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Answer: (B) $250$

By induction

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

. Adding

k=1,\dots,20

: the diagonal entries each become

20

, and the upper-right entry is

1+2+\dots+20=210

, the lower-left entry stays

0

. So

B=\begin{bmatrix}20&210\\0&20\end{bmatrix}

and the sum of its entries is

20+210+0+20=250

. Choosing

210

would count only the off-diagonal entry, and forgetting the diagonal terms is the usual error.

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