Before You Begin

Question #41254

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all pairs of $3 \times 3$ real matrices $(A, B)$ such that $A$ is an orthogonal matrix, $B$ is a nilpotent matrix of index $3$ (i.e., $B^3 = O$ and $B^2 \neq O$ ), and $A + B = I$ , where $I$ is the $3 \times 3$ identity matrix and $O$ is the null matrix. The number of elements in the set $S$ is

Options

A.
$0$
B.
$1$
C.
$3$
D.
Infinite

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Singular vs. Non-Singular Threshold|Orthogonal Matrix Conditions|Nilpotent Matrix Logic

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of orthogonal matrix properties and nilpotent matrix invariants.

The trap lies in the student's instinct to algebraically construct a strictly upper triangular matrix $B$ to satisfy $B^3 = O$ and then force $I - B$ to be orthogonal.

This leads to a messy, time-consuming system of non-linear equations.

The conceptual bypass requires recognizing that the orthogonality condition $A A^T = I$ translates to $B + B^T = B B^T$ .

By applying the trace operator—a linear invariant—students can immediately exploit the fact that the trace of a nilpotent matrix is zero, while the trace of $B B^T$ is strictly positive for any non-zero real matrix, instantly proving the set is empty.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Singular vs. Non-Singular Threshold
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free