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Question #41244

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 0 & 3 & 4 \\ 0 & 0 & 5 \\ 0 & 0 & 0 \end{pmatrix}$ and $I$ be the $3 \times 3$ identity matrix. Consider the infinite series of matrices $S = \sum_{k=1}^{\infty} (k I - A)^{-1} A ((k+1)I - A)^{-1}$ Then the sum of all the elements of the matrix $S$ is:

Options

A.
27
B.
12
C.
15
D.
30

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

Nilpotent Matrix Logic|Inverse Existence Criterion|V-N Method (Partial Fractions)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize and manipulate telescopic matrix series (V-N method) alongside the properties of nilpotent matrices.

The primary trap lies in attempting to compute the inverse $(kI - A)^{-1}$ algebraically using the adjoint method, which becomes computationally paralyzing.

Instead, students must identify the hidden difference of inverses $X^{-1} - Y^{-1} = X^{-1}(Y-X)Y^{-1}$ to collapse the infinite sum.

Furthermore, they must apply the inverse existence criterion for nilpotent matrices, recognizing that since $A^3 = 0$ , the Neumann series $(I-A)^{-1} = I + A + A^2$ terminates finitely, allowing for an exact evaluation of the boundary terms without convergence issues.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Nilpotent Matrix Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free