Math Problem Statement

A square matrix A is called nilpotent if A^k = 0 for some positive integer k. Suppose A is nilpotent. Then the value of det(A) is:

Solution

The question states that a square matrix AA is called nilpotent if Ak=0A^k = 0 for some positive integer kk, and asks for the value of det(A)\det(A).

Explanation:

If a matrix AA is nilpotent, it means that for some integer kk, raising AA to the power of kk results in the zero matrix: Ak=0A^k = 0. A fundamental property of the determinant is that if a matrix is nilpotent, its determinant must be zero.

This is because:

  • The determinant of a matrix product satisfies det(Ak)=(det(A))k\det(A^k) = (\det(A))^k.
  • If Ak=0A^k = 0, then det(Ak)=0\det(A^k) = 0, which implies that (det(A))k=0(\det(A))^k = 0.
  • Therefore, det(A)=0\det(A) = 0 (since the only number whose power can be zero is zero).

Thus, the correct answer is 0.

Would you like more details or explanations? Here are five related questions to expand on this topic:

  1. What are some real-life applications of nilpotent matrices?
  2. How do you compute the determinant of a matrix in general?
  3. Can a matrix with a determinant of 0 be invertible? Why or why not?
  4. How do eigenvalues relate to the nilpotency of a matrix?
  5. What is the significance of the minimal polynomial of a nilpotent matrix?

Tip:

In any square matrix, if its determinant is 0, the matrix is singular, meaning it has no inverse.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Theory
Nilpotent Matrices

Formulas

det(A^k) = (det(A))^k
A^k = 0 (Nilpotency condition)

Theorems

Property of Nilpotent Matrices
Determinant of Nilpotent Matrices

Suitable Grade Level

Undergraduate Linear Algebra