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Question #41764

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 11 & 24 \\ -2 & 0 \end{pmatrix}$ be two matrices. A feasible region $R$ in the $xy$ -plane is defined by the system of inequalities: $x \ge 0$ , $y \ge 0$ , and $x + 214y \le 1$ . The region $R$ is transformed into a new region $R'$ by the linear transformation $T(\mathbf{v}) = M\mathbf{v}$ , where the transformation matrix is given by $M = (A+B)^2 - (A^2+B^2)$ . What is the area of the region $R'$ ?

Options

A.
$\frac{229}{214}$
B.
$\frac{96}{107}$
C.
$\frac{24}{107}$
D.
$\frac{1429}{214}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Feasible Region Constancy|Adjoint-Determinant Relation|Determinant Scaling Property|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of linear transformations on 2D areas and the non-commutative property of matrix multiplication.

The primary trap lies in the expansion of $(A+B)^2$ .

Students accustomed to scalar algebra often incorrectly assume commutativity ( $AB = BA$ ) and simplify the transformation matrix to $M = 2AB$ .

This leads them to calculate the determinant as $\det(2AB) = 4\det(A)\det(B) = 4(2)(48) = 384$ , resulting in the trap answer $384/428 = 96/107$ .

By forcing the manual calculation of $AB + BA$ , the problem ensures that only those who respect matrix non-commutativity and correctly apply the areal scaling property ( $|\det(M)|$ ) will arrive at the correct area.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Feasible Region Constancy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free