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Question #40947

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ , $B = \begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix}$ , and $C = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ . Let $F(x) = A + xB + x^2C$ be a matrix-valued function of a real variable $x$ . If $H(x) = (F(x))^2$ , find the value of the matrix $H''(0)$ .

Options

A.
$\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$
B.
$\begin{pmatrix} 4 & 0 \\ 0 & 0 \end{pmatrix}$
C.
$\begin{pmatrix} 0 & 0 \\ 0 & 4 \end{pmatrix}$
D.
$\begin{pmatrix} 2 & -2 \\ 2 & -2 \end{pmatrix}$

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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

Algebra of Derivatives|Nilpotent Matrix Logic|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question targets a classic conceptual landmine in matrix calculus: the assumption that the scalar chain rule $d/dx[f(x)^2] = 2f(x)f'(x)$ applies to matrices.

Because matrix multiplication is non-commutative, the correct derivative of $F(x)^2$ is $F'(x)F(x) + F(x)F'(x)$ .

Applying the incorrect scalar rule leads to $H''(0) = 4AC + 2B^2$ , yielding the trap option B.

The inclusion of nilpotent matrices simplifies the arithmetic while heavily punishing students who fail to respect the strict ordering of matrix multiplication.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Algebra of Derivatives
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free