is Periodic Matrix Definition & Examples

1. is Periodic Matrix ?
`[[1,-2,-6],[-3,2,9],[2,0,-3]]`

Solution:

A square matrix `A` is called a periodic matrix, if `A^m = A` for some positive integer m.

`A`

`1`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`A×A`

`1`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`1`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`1×1-2×-3-6×2`	`1×-2-2×2-6×0`	`1×-6-2×9-6×-3`
`-3×1+2×-3+9×2`	`-3×-2+2×2+9×0`	`-3×-6+2×9+9×-3`
`2×1+0×-3-3×2`	`2×-2+0×2-3×0`	`2×-6+0×9-3×-3`

`1+6-12`	`-2-4+0`	`-6-18+18`
`-3-6+18`	`6+4+0`	`18+18-27`
`2+0-6`	`-4+0+0`	`-12+0+9`

`-5`	`-6`	`-6`
`9`	`10`	`9`
`-4`	`-4`	`-3`

`(A^2)×A`

`-5`	`-6`	`-6`
`9`	`10`	`9`
`-4`	`-4`	`-3`

`1`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`-5×1-6×-3-6×2`	`-5×-2-6×2-6×0`	`-5×-6-6×9-6×-3`
`9×1+10×-3+9×2`	`9×-2+10×2+9×0`	`9×-6+10×9+9×-3`
`-4×1-4×-3-3×2`	`-4×-2-4×2-3×0`	`-4×-6-4×9-3×-3`

`-5+18-12`	`10-12+0`	`30-54+18`
`9-30+18`	`-18+20+0`	`-54+90-27`
`-4+12-6`	`8-8+0`	`24-36+9`

`1`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

Here `A^3 = A`, so `A` is a periodic matrix of period 2

2. is Periodic Matrix ?
`[[2,-2,-6],[-3,2,9],[2,0,-3]]`

Solution:

A square matrix `A` is called a periodic matrix, if `A^m = A` for some positive integer m.

`A`

`2`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`A×A`

`2`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`2`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`2×2-2×-3-6×2`	`2×-2-2×2-6×0`	`2×-6-2×9-6×-3`
`-3×2+2×-3+9×2`	`-3×-2+2×2+9×0`	`-3×-6+2×9+9×-3`
`2×2+0×-3-3×2`	`2×-2+0×2-3×0`	`2×-6+0×9-3×-3`

`4+6-12`	`-4-4+0`	`-12-18+18`
`-6-6+18`	`6+4+0`	`18+18-27`
`4+0-6`	`-4+0+0`	`-12+0+9`

`-2`	`-8`	`-12`
`6`	`10`	`9`
`-2`	`-4`	`-3`

`(A^2)×A`

`-2`	`-8`	`-12`
`6`	`10`	`9`
`-2`	`-4`	`-3`

`2`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`-2×2-8×-3-12×2`	`-2×-2-8×2-12×0`	`-2×-6-8×9-12×-3`
`6×2+10×-3+9×2`	`6×-2+10×2+9×0`	`6×-6+10×9+9×-3`
`-2×2-4×-3-3×2`	`-2×-2-4×2-3×0`	`-2×-6-4×9-3×-3`

`-4+24-24`	`4-16+0`	`12-72+36`
`12-30+18`	`-12+20+0`	`-36+90-27`
`-4+12-6`	`4-8+0`	`12-36+9`

`-4`	`-12`	`-24`
`0`	`8`	`27`
`2`	`-4`	`-15`

`(A^3)×A`

`-4`	`-12`	`-24`
`0`	`8`	`27`
`2`	`-4`	`-15`

`2`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`-4×2-12×-3-24×2`	`-4×-2-12×2-24×0`	`-4×-6-12×9-24×-3`
`0×2+8×-3+27×2`	`0×-2+8×2+27×0`	`0×-6+8×9+27×-3`
`2×2-4×-3-15×2`	`2×-2-4×2-15×0`	`2×-6-4×9-15×-3`

`-8+36-48`	`8-24+0`	`24-108+72`
`0-24+54`	`0+16+0`	`0+72-81`
`4+12-30`	`-4-8+0`	`-12-36+45`

`-20`	`-16`	`-12`
`30`	`16`	`-9`
`-14`	`-12`	`-3`

`(A^4)×A`

`-20`	`-16`	`-12`
`30`	`16`	`-9`
`-14`	`-12`	`-3`

`2`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`-20×2-16×-3-12×2`	`-20×-2-16×2-12×0`	`-20×-6-16×9-12×-3`
`30×2+16×-3-9×2`	`30×-2+16×2-9×0`	`30×-6+16×9-9×-3`
`-14×2-12×-3-3×2`	`-14×-2-12×2-3×0`	`-14×-6-12×9-3×-3`

`-40+48-24`	`40-32+0`	`120-144+36`
`60-48-18`	`-60+32+0`	`-180+144+27`
`-28+36-6`	`28-24+0`	`84-108+9`

`-16`	`8`	`12`
`-6`	`-28`	`-9`
`2`	`4`	`-15`

`(A^5)×A`

`-16`	`8`	`12`
`-6`	`-28`	`-9`
`2`	`4`	`-15`

`2`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`-16×2+8×-3+12×2`	`-16×-2+8×2+12×0`	`-16×-6+8×9+12×-3`
`-6×2-28×-3-9×2`	`-6×-2-28×2-9×0`	`-6×-6-28×9-9×-3`
`2×2+4×-3-15×2`	`2×-2+4×2-15×0`	`2×-6+4×9-15×-3`

`-32-24+24`	`32+16+0`	`96+72-36`
`-12+84-18`	`12-56+0`	`36-252+27`
`4-12-30`	`-4+8+0`	`-12+36+45`

`-32`	`48`	`132`
`54`	`-44`	`-189`
`-38`	`4`	`69`

`(A^6)×A`

`-32`	`48`	`132`
`54`	`-44`	`-189`
`-38`	`4`	`69`

`2`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`-32×2+48×-3+132×2`	`-32×-2+48×2+132×0`	`-32×-6+48×9+132×-3`
`54×2-44×-3-189×2`	`54×-2-44×2-189×0`	`54×-6-44×9-189×-3`
`-38×2+4×-3+69×2`	`-38×-2+4×2+69×0`	`-38×-6+4×9+69×-3`

`-64-144+264`	`64+96+0`	`192+432-396`
`108+132-378`	`-108-88+0`	`-324-396+567`
`-76-12+138`	`76+8+0`	`228+36-207`

`56`	`160`	`228`
`-138`	`-196`	`-153`
`50`	`84`	`57`

`(A^7)×A`

`56`	`160`	`228`
`-138`	`-196`	`-153`
`50`	`84`	`57`

`2`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`56×2+160×-3+228×2`	`56×-2+160×2+228×0`	`56×-6+160×9+228×-3`
`-138×2-196×-3-153×2`	`-138×-2-196×2-153×0`	`-138×-6-196×9-153×-3`
`50×2+84×-3+57×2`	`50×-2+84×2+57×0`	`50×-6+84×9+57×-3`

`112-480+456`	`-112+320+0`	`-336+1440-684`
`-276+588-306`	`276-392+0`	`828-1764+459`
`100-252+114`	`-100+168+0`	`-300+756-171`

`88`	`208`	`420`
`6`	`-116`	`-477`
`-38`	`68`	`285`

`(A^8)×A`

`88`	`208`	`420`
`6`	`-116`	`-477`
`-38`	`68`	`285`

`2`	`-2`	`-6`
`-3`	`2`	`9`
`2`	`0`	`-3`

`88×2+208×-3+420×2`	`88×-2+208×2+420×0`	`88×-6+208×9+420×-3`
`6×2-116×-3-477×2`	`6×-2-116×2-477×0`	`6×-6-116×9-477×-3`
`-38×2+68×-3+285×2`	`-38×-2+68×2+285×0`	`-38×-6+68×9+285×-3`

`176-624+840`	`-176+416+0`	`-528+1872-1260`
`12+348-954`	`-12-232+0`	`-36-1044+1431`
`-76-204+570`	`76+136+0`	`228+612-855`

`392`	`240`	`84`
`-594`	`-244`	`351`
`290`	`212`	`-15`

`A` is not a periodic matrix

This material is intended as a summary. Use your textbook for detail explanation.
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1. Definition & Examples