Math Problem Statement
Solution
The question states that a square matrix is called nilpotent if for some positive integer , and asks for the value of .
Explanation:
If a matrix is nilpotent, it means that for some integer , raising to the power of results in the zero matrix: . 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 .
- If , then , which implies that .
- Therefore, (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:
- What are some real-life applications of nilpotent matrices?
- How do you compute the determinant of a matrix in general?
- Can a matrix with a determinant of 0 be invertible? Why or why not?
- How do eigenvalues relate to the nilpotency of a matrix?
- 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