Show that the given matrix is​ nilpotent, and then use this fact to find the matrix exponential eAt. A= 1 −1 1 −1

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Show that the given matrix is​ nilpotent, and then use this fact to find the matrix exponential
eAt.
 
A=
  1 −1  
1 −1
 
 
 
A nilpotent matrix is a matrix A such that
 
left parenthesis Upper A plus Upper I right parenthesis Superscript n(A+I)n
left parenthesis AA Superscript Upper T Baseline right parenthesis Superscript nAATn
left parenthesis Upper A minus Upper I right parenthesis Superscript n(A−I)n
Upper A Superscript nAn
equals
 
the identity matrix
its own transpose
the zero matrix
itself
for some positive integer n. The smallest such n for which this holds for the given matrix is
n=enter your response here​,
for which the resulting matrix is
enter your response here.
​(Use integers or fractions for any numbers in the​ expressions.)
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