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Home > Matrix & Vector > Matrix Structure > Involutary Matrix checker example
18. is Involutary Matrix example ( Enter your problem )
( Enter your problem )
  1. Definition & Examples
  2. Example-2

2. Example-2


1. is Involutary Matrix ?
`[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]`

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


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


`A×A`=
`1``0``0``0`
`0``1``0``0`
`0``0``1``0`
`0``0``0``1`
×
`1``0``0``0`
`0``1``0``0`
`0``0``1``0`
`0``0``0``1`


=
`1×1+0×0+0×0+0×0``1×0+0×1+0×0+0×0``1×0+0×0+0×1+0×0``1×0+0×0+0×0+0×1`
`0×1+1×0+0×0+0×0``0×0+1×1+0×0+0×0``0×0+1×0+0×1+0×0``0×0+1×0+0×0+0×1`
`0×1+0×0+1×0+0×0``0×0+0×1+1×0+0×0``0×0+0×0+1×1+0×0``0×0+0×0+1×0+0×1`
`0×1+0×0+0×0+1×0``0×0+0×1+0×0+1×0``0×0+0×0+0×1+1×0``0×0+0×0+0×0+1×1`


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


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



Here `A^2 = I`, so `A` is an involutary matrix 2. is Involutary Matrix ?
`[[1,2],[3,4]]`

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


`A` = 
`1``2`
`3``4`


`A×A`=
`1``2`
`3``4`
×
`1``2`
`3``4`


=
`1×1+2×3``1×2+2×4`
`3×1+4×3``3×2+4×4`


=
`1+6``2+8`
`3+12``6+16`


=
`7``10`
`15``22`



`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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