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A nilpotent matrix is a square matrix A such that Aª is the zero matrix for some positive integer n. What can you say about det A? Why ?

A nilpotent matrix is a square matrix A such that Aª is the zero matrix for some positive integer n. What can you say about det A? Why ?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A *nilpotent matrix* is a square matrix \( A \) such that \( A^n \) is the zero matrix for some positive integer \( n \). What can you say about \(\det A\)? Why? Let \( A \) be a square matrix with integer entries, so that \( A^{-1} \) is also square with integer entries. Clearly \(\det A\) is an integer, but what integers can arise? Use the formula for the inverse of a \( 2 \times 2 \) matrix to write five \( 2 \times 2 \) examples of such matrices.
Transcribed Image Text:A *nilpotent matrix* is a square matrix \( A \) such that \( A^n \) is the zero matrix for some positive integer \( n \). What can you say about \(\det A\)? Why? Let \( A \) be a square matrix with integer entries, so that \( A^{-1} \) is also square with integer entries. Clearly \(\det A\) is an integer, but what integers can arise? Use the formula for the inverse of a \( 2 \times 2 \) matrix to write five \( 2 \times 2 \) examples of such matrices.
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