JEE Advanced 2025 Paper 2 · Mathematics · Matrices and Determinants · hard

Let N=beginpmatrix0&2&1\\0&0&2\\0&0&0endpmatrix and A=I+N. If ak denotes the element in the first row and third column of A^k, then Σk=1^10ak equals

Single correct option

Let

N=\begin{pmatrix}0&2&1\\0&0&2\\0&0&0\end{pmatrix}

and

A=I+N

. If

a_k

denotes the element in the first row and third column of

A^k

, then

\displaystyle\sum_{k=1}^{10}a_k

equals

Show answer and solution

Answer: (A) $715$

N^3=O

, so by the binomial theorem (as

I

and

N

commute)

A^k=I+kN+\binom{k}{2}N^2

. Here

N^2=\begin{pmatrix}0&0&4\\0&0&0\\0&0&0\end{pmatrix}

. Therefore

a_k=k\cdot1+\binom k2\cdot4

. Then

\sum_{k=1}^{10}a_k=\sum k+4\sum\binom k2=55+4\binom{11}{3}=55+4\cdot165=715

(confirmed by direct matrix powers). Omitting the factor

4

gives

220

, the other options come from similar slips.

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