Let `A=[a_("ij")]` be a matrix of order 2 where `a_("ij") in {-1, 0, 1}` and adj. `A=-A`. If det. `(A)=-1`, then the number of such matrices is ______ .

Answer

12

Explanation

adj. `A=-A`
`implies A.` adj. `A=-A^(2)=|A|I=-I`
`implies A^(2)=I`
Let `A=[(a,b),(c,d)]`
`implies A^(2)=[(a^(2)+bc,(a+d)b),((a+b)c,d^(2)+bc)]=[(1,0),(0,1)]`
on comparing both sides, we get
`a^(2)+bc=1, (a+d)b=0, (a+d)c=0, d^(2)+bc=1`
Case I : When `(a+d) ne 0`
`implies b=0=c` and `a=1, d=1`
or `a=-1, d=-1`
`:. A=[(1,0),(0,1)]` or `[(-1,0),(0,-1)]`
But both are rejected as det. `A=-1` (given)
Case II : When `(a+d)=0 implies d=-a`
(i) If `a=1, d=-1 implies bc =0`
For `b=0, c` can be `-1, 0, 1`
For `b=1, c` can be 0 only.
For `b=-1, c` can be 0 only.
So, 5 matrices are possible.
(ii) If `a=-1, d=1`
`implies bc=0`
For `b=0, c=-1, 0, 1`.
For `b=1, c=0` only.
For `b=-1, c=0` only.
So, 5 matrices are possible.
(iii) If `a=0, d=0`
`implies bc=1`
`:. A=[(0,1),(1,0)]` or `A=[(0,-1),(-1,0)]`
So, 2 matrices are possible.
Therefore, total number of matrices is 12.