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JEE Mathematics 2023 Question with Solution

Let A=[012103100]A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 0 & 3 \\ 1 & 0 & 0 \end{bmatrix}, where a,c∈Ra, c \in \mathbb{R}. If An=AA^n = A and the positive value of aa belongs to the interval (n−1,n](n-1, n], where n∈Nn \in \mathbb{N}, then nn is equal to:

  • A

    22

  • B

    33

  • C

    11

  • D

    44

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given:

A=[012103100]A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 0 & 3 \\ 1 & 0 & 0 \end{bmatrix}

Find: The value of nn.

From the extracted solution, the matrix powers are computed as follows:

A2=A×A=[236329236]A^2 = A \times A = \begin{bmatrix} 2 & 3 & 6 \\ 3 & 2 & 9 \\ 2 & 3 & 6 \end{bmatrix} A3=A×A2=[236329236]A^3 = A \times A^2 = \begin{bmatrix} 2 & 3 & 6 \\ 3 & 2 & 9 \\ 2 & 3 & 6 \end{bmatrix}

Thus, the working concludes that

A2=A3A^2 = A^3

and hence the solution states that n=3n = 3.

Therefore, the correct option is B.

Answer from the solution

The solution explicitly computes A2A^2 and A3A^3 and then concludes that the required value is 33. The answer key says 2, but as per the extraction rule, the solution is treated.

Therefore, the final extracted answer is B.

Common mistakes

  • A common mistake is to trust the answer key key without checking the worked solution. Here, the extracted solution concludes n=3n = 3, so the answer must be taken from the solution.

  • Another mistake is to confuse the matrix entries with the variable symbols mentioned in the statement. Read carefully which quantity is actually being used in the condition before selecting the option.

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