If a, b, c and d are real numbers such that a^2+b^2+c^2+d^2=1 and A= [ array c a+i b c+i d -c+i d a-i b array ] , then A⁻¹ equals to

Step-by-step solution

Given, a^2+b^2+c^2+d^2=1 and A= [ array cc a+i b & c+i d -c+i d & a-i b array ] aligned Now |A| & =(a+i b)(a-i b) & -(c+i d)(-c+i d) & =a^2-(i b)^2- [(i d)^2-(c)^2 ] & =a^2+b^2- [-d^2-c^2 ] & =a^2+b^2+d^2+c^2 & =1 A⁻¹ & = 1 |A| [ array cc a-i b & -(c+i d) -(-c+i d) & a+i b array ] aligned [from Eq. (i)] A⁻¹= 1 |A| [ array cc a-i b & -(c+i d) -(-c+i d) & a+i b array ] aligned & = 1 1 [ array cc a-i b & -c-i d c-i d & a+i b array ] & = [ array cc a-i b & -c-i d c-i d & a+i b array ] aligned