InterviewSolution
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InterviewSolution
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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 ______ . |
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Answer» Correct Answer - 12 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. |
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