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Question #40575

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = [a_{ij}]$ be a $4 \times 4$ matrix where $a_{ij}$ represents the number of ways to select $2$ distinct objects from a pool of $i+j$ distinct objects. Let $B = \text{adj}(\text{adj}(A))$ . Find the value of $\det(B^3 + 2B^2 + 5I_4)$ , where $I_4$ is the $4 \times 4$ identity matrix.

Options

A.
$0$
B.
$25$
C.
$125$
D.
$625$

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

Selection vs Arrangement|Adjoint-Determinant Relationship|Determinant Product Rule

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of combinatorial matrix construction and advanced adjoint properties.

Students often fail to recognize that a matrix formed by a polynomial function of $(i+j)$ of degree $d$ has a maximum rank of $d+1$ .

Here, the degree is 2, so the rank is at most 3, making the $4 \times 4$ matrix singular ( $|A|=0$ ).

The primary trap lies in the adjoint evaluation: students might incorrectly assume that $|A|=0$ only implies $|B|=0$ and attempt to evaluate the determinant of the polynomial matrix using eigenvalues, missing the stronger matrix identity $\text{adj}(\text{adj}(A)) = |A|^{n-2}A = O$ for $n \ge 3$ .

Recognizing $B$ as the null matrix trivially collapses the complex polynomial into a scalar matrix determinant.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Selection vs Arrangement
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free