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Question #40517

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix where $a_{ij}$ is the number of integers $k$ satisfying the inequality $k^2 - (i+2j)k + 2ij \le 0$ for $1 \le i, j \le 3$ . Let $B$ be the $3 \times 3$ matrix such that for any $3 \times 3$ matrix $Y$ , the product $BY$ represents the matrix obtained by first swapping the first and second rows of $Y$ , and then swapping the second and third rows of the resulting matrix. Consider the system of linear equations $M X = C$ , where $M = (A+B)(A-B) - A^2 + B^2$ and $C = \begin{pmatrix} -4 \\ -2 \\ 0 \end{pmatrix}$ . If $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ is the unique solution to the system, find the value of $x + y + z$ .

Options

A.
2
B.
-4
C.
0
D.
4

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

Sign Flipping Rule|System of Linear Equations (Cramer/Inverse)|Matrix Multiplication Non-Commutativity|Characteristic Equation

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance against the 'Non-Commutativity' trap in matrix algebra, combined with the 'Sign Flipping Rule' of determinants and row operations.

Students often mistakenly assume $(A+B)(A-B) = A^2 - B^2$ , which would lead to $M = 0$ and the false conclusion that the system has no unique solution.

By correctly expanding the expression to $M = BA - AB$ , the student must then construct $A$ from the quadratic inequality (yielding a specific integer matrix) and $B$ from the described row operations (yielding a cyclic permutation matrix).

The row operations implicitly test the sign flipping rule, as $B$ is formed by two row swaps ( $\det(B)=1$ ).

Finally, applying Cramer's rule or Gaussian elimination to $MX = C$ yields the correct sum of the solution components.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Sign Flipping Rule
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free