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Find Is matrix [[-5,-8,0],[3,5,0],[1,2,-1]]

Solution:
Your problem `->` Is matrix [[-5,-8,0],[3,5,0],[1,2,-1]]




A matrix, in which number of rows and number of columns are equal, is called a square matrix.

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


The number of rows(3) and number of columns(3) are equal, So `A` is a square matrix



A square matrix `A`, such that `|A| != 0`, is called nonsingular matrix.


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


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


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


`=-5 xx (5 × (-1) - 0 × 2) +8 xx (3 × (-1) - 0 × 1) +0 xx (3 × 2 - 5 × 1)`

`=-5 xx (-5 +0) +8 xx (-3 +0) +0 xx (6 -5)`

`=-5 xx (-5) +8 xx (-3) +0 xx (1)`

`= 25 -24 +0`

`=1`


Here, `|A| != 0`, so `A` is nonsingular matrix



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



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


`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`



`(A^2)×A`=
`1``0``0`
`0``1``0`
`0``0``1`
×
`-5``-8``0`
`3``5``0`
`1``2``-1`


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


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


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



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








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