Paragraph: Let A= ( array ccc 0 & ^2 & ^2 ^2 & 0 & ^2 ^2 & ^2 & 0 array ) and A=P+Q where P is symmetric matrix and Q is skew-symmetric matrix. Question: What is the minimum value of determinant of A ?

Step-by-step solution

aligned & |A|= [ array ccc 0 & ^2 & ^2 ^2 & 0 & ^2 ^2 & ^2 & 0 array ] & =0- ^2 (0- ^4 )+ ^2 ( ^4 ) & = ^6 + ^6 & = ( ^2 + ^2 ) ( ^4 + ^4 - ^2 ^2 ) & = ( ^2 + ^2 )^2-3 ^2 ^2 & =1- 3 4 ^2 2 = 1 4 [4-3 (1- 4 ) 2 ] & = 1 8 [5+3 4 ] aligned Minimum value of 4 is -1 . |A|_ = 1 8 [5-3]= 2 8 = 1 4