If `A = {:((1,-1,2),(3,0,-2),(1,0,3)):}`, verify that A (adj A) = `|A|* I`.

Explanation

We have `|{:(,1,-1,2),(,3,0,-2),(,1,0,3):}|`
` therefore" "|A| = |{:(,1,-1,2),(,3,0,-2),(,1,0,3):}|`
` = 1 (0+0) +1(9+2)+2(0-0)`
` = 0 + 11 + 0`
` = 11`
Now , the cofactors of A are .
`A_(11) = (-1)^(1+1) M_(11) = |{:(,0,-2),(,0,3):}|`
` = 0 + 0 = 0`
`A_(12) = (-1)^(1+3) M_(13) = - |{:(, 3,-2),(,1,3):}|`
= 0 - 0 = 0
`A_(21) = (-1)^(1+3)M_(21) = - |{:(,-1,2),(,0,3):}|`
` = - (-3 - 0 ) = 3`
`A_(23) = (-1)^(2+3) M_(23) = - |{:(, 1,-1),(,1,0):}|`
` = - (0+1) = - 1`
`A_(31) = (-1)^(3+1)M_(31) = - |{:(,-1,2),(,0,-2):}|`
` = 2- 0 = 2`
`A_(32) = (-1)^(3+2)M_(32) = |{:(,1,2),(,3,-2):}|`
` = -(-2-6) = 8`
`A_(33) = (-1)^(3+3)M_(33) = |{:(,1,-1),(,3,0):}|`
= 0 + 3 = 3
Hence the confoctor matrix .
` ={{:(,A_(11),A_(12),A_(13)),(,A_(21),A_(22),A_(23)),(,A_(31),A_(32),A_(33)):}}`
` = [{:(0,-11,0),(3,1,-1),(2,8,3):}]`

`therefore " "adj.A = [{:(,0,-11,0),(,3,1,-1),(,2,8,3):}]`
`= [{:(0,3,2),(-11,1,8),(0,-1,3):}]`
`therefore" "A(adj. A) = [{:(,1,-1,2),(,3,0,-2),(,1,0,3):}][{:(,0,3,2),(,-11,1,8),(,0,-1,3):}]`
` = [{:(, 0+11+0,3-1-2, 2-8+6),(,0+0-0,9+0+2, 6+0-6),(,0+0+0,3+0-3,2+0+9):}]`
`=[{:(,11,0,0),(,0,11,0),(,0,0,11):}]" ".......(1)`
`|A| . I = 11[{:(,1,0,0),(,0,1,0),(,0,0,1):}]`
` = [{:(,11,0,0),(,0,11,0),(,0,0,11):}]" "......(2)`
From equation (1) and (2)
A(adj. A) = |A| . I