JEE Advanced · Mathematics · 19. Determinants

Let P be a matrix of order $3 \times 3$ such that all the entries in P are from the set $\{- 1, 0, 1\}$ . Then, the maximum possible value of the determinant of P is _______.

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Answer & Solution

Correct Answer

(C) 4

Step-by-step Solution

Detailed explanation

∆ = |\begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix}| = \begin{matrix} \underset{⏟}{(a_{1} b_{2} c_{3} + a_{2} b_{3} c_{1} + a_{3} b_{1} c_{2})} \\ x \end{matrix} - \begin{matrix} \underset{⏟}{(a_{3} b_{2} c_{1} + a_{2} b_{1} c_{3} + a_{1} b_{3} c_{2})} \\ y \end{matrix}

Now if

x \leq 3

and

y \geq - 3

The

∆

can be maximum 6 but it is not possible
because

x = 3 \Rightarrow

each term of

x = 1

And

y = -3 \Rightarrow

each term of

y = - 1

\Rightarrow

\prod_{i = 1}^{3} a_{i} b_{i} c_{i} = 1

and

\prod_{i = 1}^{3} a_{i} b_{i} c_{i} = - 1

, which is a contradiction.
Now for value to be

5

one the terms must be zero but that will make two terms zero which means answer cannot be

5

. Next possibility is

4

which can be obtained as

|\begin{matrix} 1 & 1 & 1 \\ - 1 & 1 & 1 \\ 1 & - 1 & 1 \end{matrix}| = 1 (1 + 1) - 1 (- 1 - 1) + 1 (1 - 1) = 4

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let P be a matrix of order 3×3 such that all the entries in P are from the set -1, 0, 1 . Then, the maximum possible value of the determinant of P is _______.

Step-by-step Solution

Let P be a matrix of order $3 \times 3$ such that all the entries in P are from the set $\{- 1, 0, 1\}$ . Then, the maximum possible value of the determinant of P is _______.