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Question #41238

Difficulty: 8/10 | 🟢 Free

The Question

Let $m$ and $n$ be the order and degree, respectively, of the differential equation $\frac{d^3y}{dx^3} = \left( 1 + \left(\frac{d^2y}{dx^2}\right)^4 \right)^{\frac{1}{3}}$ Let $A$ be an $m \times m$ matrix such that $A^m = O$ and $A^{m-1} \neq O$ , where $O$ is the null matrix. If $M = (I + A)^{-n} + (I - A)^{-n}$ , then the value of $\det(n M)$ is

Options

A.
8
B.
54
C.
216
D.
729

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Inversion and Determinant Product|Nilpotent Matrix Logic|Order and Degree Definition

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of differential equation properties and nilpotent matrix algebra.

The primary trap lies in the inversion of $(I \pm A)$ .

Students often erroneously associate the singularity of a nilpotent matrix ( $\det(A)=0$ ) with the non-invertibility of its transformations, or they fail to truncate the infinite geometric/binomial series $(I+A)^{-n}$ using $A^m=O$ .

Furthermore, a secondary lethal collision occurs in the determinant evaluation: extracting a scalar $k$ from an $m \times m$ matrix yields $k^m \det(X)$ .

Students who forget the matrix order $m=3$ will incorrectly compute $\det(6(I+6A^2))$ as $6 \times 1 = 6$ or $27 \times 2 = 54$ , falling for the distractor options.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Inversion and Determinant Product
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free