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Question #40373

Difficulty: 8/10 | 🟢 Free

The Question

Let $B = \begin{bmatrix} x & 1 & 2 \\ 1 & x & -1 \\ 2 & -1 & x \end{bmatrix}$ and $A = [a_{ij}]$ be a $3 \times 3$ matrix where $a_{ij}$ is the minor of the element $b_{ij}$ in $B$ . A mapping $f: \mathbb{R}^3 \to \mathbb{R}^3$ is defined by $f(V) = AV$ , where $V$ is a $3 \times 1$ column vector. If $f$ is not a one-to-one mapping, then the sum of the squares of all possible real values of $x$ is

Options

A.
12
B.
16
C.
24
D.
36

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

Function Definition (The 'One-to-One' Constraint)|Singular vs Non-Singular Matrices|Minors and Cofactors Sign Convention

Why This Question Is Tricky

Common Mistakes

This question tests the conceptual link between the injectivity of a linear transformation and matrix singularity, combined with a classic trap involving minors and cofactors.

Students often attempt to manually construct the matrix of minors $A$ and compute its determinant, leading to a convoluted degree-6 polynomial and inevitable sign errors.

The intended logical bypass is recognizing that the minor matrix $A$ is related to the cofactor matrix $C$ via $A = DCD$ (where $D$ is a diagonal matrix of alternating signs), meaning $\det(A) = \det(C) = (\det B)^2$ .

This elegantly reduces the problem to simply finding the roots of $\det(B) = 0$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Function Definition (The 'One-to-One' Constraint)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free