NIMCET 2025 — Mathematics PYQ

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Question

NIMCET 2025 — Mathematics PYQ

NIMCET | Mathematics | 2025

Consider the matrix $B = \begin{pmatrix} -1 & -1 & 2 \\ 0 & -1 & -1 \\ 0 & 0 & -1 \end{pmatrix}$ . The sum of all the entries of the matrix $B^{19}$ is:

Choose the correct answer:

Explanation

Solution

Step 1: Matrix Decomposition

B = -I + N = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix} + \begin{pmatrix} 0 & -1 & 2 \\ 0 & 0 & -1 \\ 0 & 0 & 0 \end{pmatrix}

Where $N = \begin{pmatrix} 0 & -1 & 2 \\ 0 & 0 & -1 \\ 0 & 0 & 0 \end{pmatrix}$ .

Step 2: Powers of $N$

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

N^3 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} = O

Step 3: Expansion of $B^{19}$

Using $( -I + N )^{19} = - ( I - N )^{19}$ :

B^{19} = - \left[ \binom{19}{0}I - \binom{19}{1}N + \binom{19}{2}N^2 \right]

B^{19} = - [ I - 19N + 171N^2 ]

Step 4: Substitute Values

B^{19} = - \left[ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} - \begin{pmatrix} 0 & -19 & 38 \\ 0 & 0 & -19 \\ 0 & 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 & 171 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \right]

B^{19} = - \begin{pmatrix} 1 & 19 & 133 \\ 0 & 1 & 19 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} -1 & -19 & -133 \\ 0 & -1 & -19 \\ 0 & 0 & -1 \end{pmatrix}

Step 5: Sum of Entries

\text{Sum} = (-1) + (-19) + (-133) + (0) + (-1) + (-19) + (0) + (0) + (-1)

\text{Sum} = -174

Final Answer:

The sum of all entries is $-174$ . (Option A)