Let A be a 3 3 matrix such that X^T A X=O for all nonzero 3 1 matrices X= [ array l x y z array ] . If A [ array l 1 1 1 array ]= [ array c 1 4 -5 array ], A [ array l 1 2 1 array ]= [ array c 0 4 -8 array ] , and det ( adj (2( A + 1 )))-2^ 3^ 5^ , , , N , then ^2+ ^2+ ^2 is_____.

Step-by-step solution

aligned & X^ A X=0 & (x y z) [ array lll a₁ & a₂ & a₃ b₁ & b₂ & b₃ c₁ & c₂ & c₃ array ] [ array l x y z array ]=0 & (x y z) [ array l a₁ x+a₂ y+a₃ z b₁ x+b₂ y+b₃ z c₁ x+c₂ y+c₃ z array ]=0 & x (a₁ x+a₂ y+a₃ z )+y (b₁ x+b₂ y+b₃ z ) & +z (c₁ x+c₂ y+c₃ z ) & a₁=0, b₂=0, c₃=0 & a₂+b₁=0, a₃+c₁=0, b₃+c₂=0 & A= skew symmetric matrix & A= [ array ccc 0 & x & y -x & 0 & z -y & -z & 0 array ] ; A [ array l 1 1 1 array ]= [ array c 1 4 -5 array ] & [ array ccc 0 & x & y -x & 0 & z -y & -z & 0 array ] [ array l 1 1 1 array ]=