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Question #41204

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{pmatrix}$ . Consider the system of linear equations given by $(A^{2024} - I) \mathbf{x} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ , where $I$ is the $3 \times 3$ identity matrix and $\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ . If this system has infinitely many solutions, which of the following conditions must be true?

Options

A.
$a + b = 0$ and $a - c = 0$
B.
$a - b = 0$ and $a + c = 0$
C.
$a = 0, b = 0,$ and $c = 0$
D.
$2a + b + c = 0$ and $a - b = 0$

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

Idempotent and Involutory Properties|Characteristic Equation|Singular vs. Non-Singular Threshold

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to identify and apply the idempotent property of a matrix ( $A^2 = A$ ) to simplify high-power matrix expressions.

The core trap lies in the 'Singular Threshold': students often confuse idempotent matrices with involutory matrices ( $A^2 = I$ ).

If a student incorrectly assumes $A^2 = I$ , they will conclude $A^{2024} - I = 0$ , leading to the erroneous trivial condition $a=b=c=0$ .

Furthermore, by asking for the condition for infinitely many solutions, it forces students to evaluate the column space of the singular matrix $A - I$ , penalizing those who blindly attempt to find an inverse without checking the determinant.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Idempotent and Involutory Properties
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free