Before You Begin

Question #40656

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ real invertible matrix and let $B = (AA^T)^2 - 2AA^T$ . If $\text{Tr}(B) = 2$ , $\text{Tr}(B^2) = 10$ , and $\text{Tr}(B^3) = 26$ , which of the following is the value of $\det(AA^T + I)$ ?

Options

A.
12
B.
24
C.
36
D.
48

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

Expansion of Products of Binomials|Characteristic Equation|Transpose of Product|Linear Independence vs Dependence

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Newton-Girard formulae with the spectral properties of symmetric positive-definite matrices.

The primary trap lies in attempting to solve the matrix equation $B = (AA^T)^2 - 2AA^T$ algebraically or assuming $B$ is positive definite.

Students must recognize that the traces of powers of $B$ uniquely determine its characteristic polynomial via Newton's sums, revealing its eigenvalues.

The 'Lethal Collision' occurs when mapping the eigenvalues of $B$ back to $AA^T$ : students often forget that for an invertible matrix $A$ , the product $AA^T$ is strictly positive definite.

Failing to apply the constraint $\mu > 0$ leads to incorrect eigenvalues (like $0$ or $-1$ ) for $AA^T$ , resulting in a completely wrong determinant.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Expansion of Products of Binomials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free