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Question #41786

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ and $B$ be $3 \times 3$ matrices with real entries. Define the matrix $C = AB - BA + 5I_3$ , where $I_3$ is the $3 \times 3$ identity matrix. Let the matrix $C$ be given by $C = \begin{pmatrix} \alpha & 2 & 1 \\ -1 & \beta & 4 \\ 3 & 0 & \gamma \end{pmatrix}$ Consider the following system of linear equations in variables $x, y, z$ : $x + 2y + 3z = \alpha$ $2x + 3y + 5z = \beta$ $7x + 10y + 17z = \gamma$ If this system of equations is consistent, what is the value of $\beta$ ?

Options

A.
2
B.
3
C.
4
D.
5

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

Inconsistency Detection|Matrix Trace Properties|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of matrix trace properties and the conditions for consistency in a system of linear equations.

A common trap is for students to incorrectly assume that matrix multiplication is commutative ( $AB = BA$ ), which would lead them to conclude that $C = 5I_3$ and thus $\beta = 5$ .

However, this assumption makes the linear system inconsistent, contradicting the problem statement.

The correct approach requires recognizing that the trace of the commutator $[A, B]$ is always zero, yielding $\alpha + \beta + \gamma = 15$ .

By identifying the linear dependence in the system's coefficients ( $4L_2 - L_1 = L_3$ ), students can derive the consistency constraint $\alpha + \gamma = 4\beta$ , which elegantly resolves to the correct value of $\beta$ without needing to know the individual matrices.

Quick Facts About This Question

Question Information

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