If `a=3+2i,b=3-2i` show that `a^(2)+ab+b^(2)=23`

Solve the following: If a = (-1 + sqrt(3)i) / 2 , b = (-1 - sqrt(3)i) / 2 , show that a^2 = b and b ^2 = a

If a= frac{-1+sqrt3 i}{2} , b= frac{-1-sqrt3 i}{2} , then show that a^2 =b and b^2 =a

If A= ((2,3),(-1,2)) .Show that A^(2)-4A+7I=0 whwre I is the unit matrix of order 2.

If a^(2) + b^(2) = 23ab , show that : "log" (a+b)/(5) = (1)/(2) (log a + log b) .

If (1+i)(2+i)(3+i)...(n+i)=a+ib show that, 2*5*10...(n^(2)+1)=a^(2)+b^(2).

If A={:((2,3),(6,5)):}and B={:((3,7),(4,0)):} , express (A+B)^(2) as a matrix and show that, (A+B)^(2)=A^(2)+AB+BA+B^(2) .

If A = {:[(7,5),(2,3)] , show that A^2 - 5A + 7I = O .

If A(-2,1),B(2,3) and C(-2,-4) be the vertices of a triangle ABC , show that tan B=(2)/(3)

If A=([3,23,3]) and B=([1,-(2)/(3)-1,1]) show that AB=I_(2) where I_(2) is the unit matrix of order 2

If a = 2 and b = 3, find the value of (i) a + b (ii) a^(2) + ab (iii) ab - a^(2) (iv) 2a - 3b (v) 5a^(2) - 2ab (vi) a^(3) - b^(3)