Before You Begin

Question #40578

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $3 \times 3$ matrices $A$ of the form $A = \begin{bmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{bmatrix}$ where $a, b, c, d, e, f$ are non-negative integers such that $a+b+c+d+e+f = 10$ . Find the number of matrices $A \in S$ such that the matrix $A^2 + \text{adj}(A)$ is singular.

Options

A.
858
B.
660
C.
726
D.
2211

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

Distributing Identical Items (Stars and Bars)|Inversion and Determinant Product|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of matrix algebra and combinatorics (Stars and Bars).

The 'Lethal Collision' occurs when students analyze the singularity condition of $B = A^2 + \text{adj}(A)$ .

A common trap is incorrectly assuming that $\det(B) = 0$ implies only one of the diagonal elements of $A$ must be zero, leading to a massive overcounting error (2211).

Furthermore, even if students correctly deduce that at least two diagonal elements must be zero, they frequently fail to apply the Principle of Inclusion-Exclusion correctly, either double-counting the intersection (yielding 858) or excluding the all-zero case entirely (yielding 660).

The problem demands rigorous algebraic deduction followed by precise restricted partition counting.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Distributing Identical Items (Stars and Bars)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free