1. Auto detect the matrix type ?
`[[-5,-8,0],[3,5,0],[1,2,-1]]`

Solution:

---

`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`

=

`-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`

=

`-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`

=

`-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