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Home > Matrix & Vector calculators > is Involutary Matrix example
18. is Involutary 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
17. is Nilpotent Matrix
(Previous method)		2. Example-2
(Next example)

1. Definition & Examples


1. is Involutary Matrix ?
`[[-5,-8,0],[3,5,0],[1,2,-1]]`

Solution:
A square matrix `A` is called an involutary matrix, if `A^2 = I` where `I` is the identity matrix.


`A` = 
`-5``-8``0`
`3``5``0`
`1``2``-1`


`A×A`=
`-5``-8``0`
`3``5``0`
`1``2``-1`
×
`-5``-8``0`
`3``5``0`
`1``2``-1`


=
`-5×-5-8×3+0×1``-5×-8-8×5+0×2``-5×0-8×0+0×-1`
`3×-5+5×3+0×1``3×-8+5×5+0×2``3×0+5×0+0×-1`
`1×-5+2×3-1×1``1×-8+2×5-1×2``1×0+2×0-1×-1`


=
`25-24+0``40-40+0``0+0+0`
`-15+15+0``-24+25+0``0+0+0`
`-5+6-1``-8+10-2``0+0+1`


=
`1``0``0`
`0``1``0`
`0``0``1`



Here `A^2 = I`, so `A` is an involutary matrix

2. is Involutary Matrix ?
`[[2,-8,0],[3,5,0],[1,2,-1]]`

Solution:
A square matrix `A` is called an involutary matrix, if `A^2 = I` where `I` is the identity matrix.


`A` = 
`2``-8``0`
`3``5``0`
`1``2``-1`


`A×A`=
`2``-8``0`
`3``5``0`
`1``2``-1`
×
`2``-8``0`
`3``5``0`
`1``2``-1`


=
`2×2-8×3+0×1``2×-8-8×5+0×2``2×0-8×0+0×-1`
`3×2+5×3+0×1``3×-8+5×5+0×2``3×0+5×0+0×-1`
`1×2+2×3-1×1``1×-8+2×5-1×2``1×0+2×0-1×-1`


=
`4-24+0``-16-40+0``0+0+0`
`6+15+0``-24+25+0``0+0+0`
`2+6-1``-8+10-2``0+0+1`


=
`-20``-56``0`
`21``1``0`
`7``0``1`



`A^2 != I`, so `A` is not an involutary matrix



This material is intended as a summary. Use your textbook for detail explanation.
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17. is Nilpotent Matrix
(Previous method)		2. Example-2
(Next example)


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