We use cookies to improve your experience on our site and to show you relevant advertising. By browsing this website, you agree to our use of cookies. Learn more   

AtoZmath.com
Support us
(New) All problem can be solved using search box
I want to sell my website www.AtoZmath.com with complete code
Home What's new College Algebra Games Feedback About us
Algebra Matrix & Vector Numerical Methods Statistical Methods Operation Research Word Problems Calculus Geometry Pre-Algebra


Home > Matrix & Vector calculators > is Nilpotent Matrix example
17. is Nilpotent Matrix example ( Enter your problem )
( Enter your problem )
  1. Definition & Examples
  2. Example-2
 		Other related methods
  1. is Row Matrix
  2. is Column Matrix
  3. is Square Matrix
  4. is Horizontal Matrix
  5. is Vertical Matrix
  6. is Diagonal Matrix
  7. is Identity Matrix
  8. is Scalar Matrix
  9. is Null Matrix
  10. is Lower Triangle Matrix
  11. is Upper Triangle Matrix
  12. is Orthogonal Matrix
  13. is Singular Matrix
  14. is Nonsingular Matrix
  15. is Symmetric Matrix
  16. is Skew Symmetric Matrix
  17. is Nilpotent Matrix
  18. is Involutary Matrix
  19. is Idempotent Matrix
  20. is Periodic Matrix
  21. is Positive Definite Matrix
  22. is Negative Definite Matrix
  23. is Derogatory Matrix
  24. is Diagonally Dominant Matrix
  25. is Strictly Diagonally Dominant Matrix
  26. Auto detect the matrix type
1. Definition & Examples
(Previous example)		18. is Involutary Matrix
(Next method)

2. Example-2


2. is Nilpotent Matrix ?
`[[2,2,3],[1,2,3],[-1,-2,-3]]`

Solution:
A square matrix `A` is called a nilpotent matrix, if `A^m = 0` for some positive integer m.


`A` = 
`2``2``3`
`1``2``3`
`-1``-2``-3`


`A×A`=
`2``2``3`
`1``2``3`
`-1``-2``-3`
×
`2``2``3`
`1``2``3`
`-1``-2``-3`


=
`2×2+2×1+3×-1``2×2+2×2+3×-2``2×3+2×3+3×-3`
`1×2+2×1+3×-1``1×2+2×2+3×-2``1×3+2×3+3×-3`
`-1×2-2×1-3×-1``-1×2-2×2-3×-2``-1×3-2×3-3×-3`


=
`4+2-3``4+4-6``6+6-9`
`2+2-3``2+4-6``3+6-9`
`-2-2+3``-2-4+6``-3-6+9`


=
`3``2``3`
`1``0``0`
`-1``0``0`



`(A^2)×A`=
`3``2``3`
`1``0``0`
`-1``0``0`
×
`2``2``3`
`1``2``3`
`-1``-2``-3`


=
`3×2+2×1+3×-1``3×2+2×2+3×-2``3×3+2×3+3×-3`
`1×2+0×1+0×-1``1×2+0×2+0×-2``1×3+0×3+0×-3`
`-1×2+0×1+0×-1``-1×2+0×2+0×-2``-1×3+0×3+0×-3`


=
`6+2-3``6+4-6``9+6-9`
`2+0+0``2+0+0``3+0+0`
`-2+0+0``-2+0+0``-3+0+0`


=
`5``4``6`
`2``2``3`
`-2``-2``-3`



`(A^3)×A`=
`5``4``6`
`2``2``3`
`-2``-2``-3`
×
`2``2``3`
`1``2``3`
`-1``-2``-3`


=
`5×2+4×1+6×-1``5×2+4×2+6×-2``5×3+4×3+6×-3`
`2×2+2×1+3×-1``2×2+2×2+3×-2``2×3+2×3+3×-3`
`-2×2-2×1-3×-1``-2×2-2×2-3×-2``-2×3-2×3-3×-3`


=
`10+4-6``10+8-12``15+12-18`
`4+2-3``4+4-6``6+6-9`
`-4-2+3``-4-4+6``-6-6+9`


=
`8``6``9`
`3``2``3`
`-3``-2``-3`



`(A^4)×A`=
`8``6``9`
`3``2``3`
`-3``-2``-3`
×
`2``2``3`
`1``2``3`
`-1``-2``-3`


=
`8×2+6×1+9×-1``8×2+6×2+9×-2``8×3+6×3+9×-3`
`3×2+2×1+3×-1``3×2+2×2+3×-2``3×3+2×3+3×-3`
`-3×2-2×1-3×-1``-3×2-2×2-3×-2``-3×3-2×3-3×-3`


=
`16+6-9``16+12-18``24+18-27`
`6+2-3``6+4-6``9+6-9`
`-6-2+3``-6-4+6``-9-6+9`


=
`13``10``15`
`5``4``6`
`-5``-4``-6`



`(A^5)×A`=
`13``10``15`
`5``4``6`
`-5``-4``-6`
×
`2``2``3`
`1``2``3`
`-1``-2``-3`


=
`13×2+10×1+15×-1``13×2+10×2+15×-2``13×3+10×3+15×-3`
`5×2+4×1+6×-1``5×2+4×2+6×-2``5×3+4×3+6×-3`
`-5×2-4×1-6×-1``-5×2-4×2-6×-2``-5×3-4×3-6×-3`


=
`26+10-15``26+20-30``39+30-45`
`10+4-6``10+8-12``15+12-18`
`-10-4+6``-10-8+12``-15-12+18`


=
`21``16``24`
`8``6``9`
`-8``-6``-9`



`(A^6)×A`=
`21``16``24`
`8``6``9`
`-8``-6``-9`
×
`2``2``3`
`1``2``3`
`-1``-2``-3`


=
`21×2+16×1+24×-1``21×2+16×2+24×-2``21×3+16×3+24×-3`
`8×2+6×1+9×-1``8×2+6×2+9×-2``8×3+6×3+9×-3`
`-8×2-6×1-9×-1``-8×2-6×2-9×-2``-8×3-6×3-9×-3`


=
`42+16-24``42+32-48``63+48-72`
`16+6-9``16+12-18``24+18-27`
`-16-6+9``-16-12+18``-24-18+27`


=
`34``26``39`
`13``10``15`
`-13``-10``-15`



`(A^7)×A`=
`34``26``39`
`13``10``15`
`-13``-10``-15`
×
`2``2``3`
`1``2``3`
`-1``-2``-3`


=
`34×2+26×1+39×-1``34×2+26×2+39×-2``34×3+26×3+39×-3`
`13×2+10×1+15×-1``13×2+10×2+15×-2``13×3+10×3+15×-3`
`-13×2-10×1-15×-1``-13×2-10×2-15×-2``-13×3-10×3-15×-3`


=
`68+26-39``68+52-78``102+78-117`
`26+10-15``26+20-30``39+30-45`
`-26-10+15``-26-20+30``-39-30+45`


=
`55``42``63`
`21``16``24`
`-21``-16``-24`



`(A^8)×A`=
`55``42``63`
`21``16``24`
`-21``-16``-24`
×
`2``2``3`
`1``2``3`
`-1``-2``-3`


=
`55×2+42×1+63×-1``55×2+42×2+63×-2``55×3+42×3+63×-3`
`21×2+16×1+24×-1``21×2+16×2+24×-2``21×3+16×3+24×-3`
`-21×2-16×1-24×-1``-21×2-16×2-24×-2``-21×3-16×3-24×-3`


=
`110+42-63``110+84-126``165+126-189`
`42+16-24``42+32-48``63+48-72`
`-42-16+24``-42-32+48``-63-48+72`


=
`89``68``102`
`34``26``39`
`-34``-26``-39`



`A` is not a nilpotent matrix



This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here


1. Definition & Examples
(Previous example)		18. is Involutary Matrix
(Next method)


Share this solution or page with your friends.


Home What's new College Algebra Games Feedback About us
 
Copyright © 2024. All rights reserved. Terms, Privacy
 
 

.