If A^T denotes the transpose of the matrix A= [ array lll 0 & 0 & a 0 & b & c d & e & f array ] , where a, b, c, d, e and f are integers such that a b d 0 , then the number of such matrices for which A⁻¹=A^T is

Step-by-step solution

aligned & A= [ array lll 0 & 0 & a 0 & b & c d & e & f array ],|A|=-a b d 0 & c₁₁=+(b f-c e), c₁₂=-(-c ~d )=c d, & c₂₁=-(-c a)=a e, c₂₂=+(-a d)=-a d, & c₃₁=+(-a b)=-a b, c₃₂ c₂₃=-(0)=0 & Adj A= [ array ccc (b f-c e) & a e & -a b c d & -a d & 0 -b d & 0 & 0 array ] aligned A⁻¹= 1 |A| ( adj A)= 1 a b d [ array ccc b f-c e & a e & -a b c d & -a d & 0 -b d & 0 & 0 array ] aligned & A^T= [ array lll 0 & 0 & d 0 & b & e a & c & f array ] & Now A⁻¹=A^T & 1 -a b d [ array ccc b f-c e & a e & -a b c d & -a d & 0 -b d & 0 &