The number of `2xx2` matrices `A`, that are there with the elements as real numbers satisfying `A+A^(T)=I` and `A A^(T)=I` is (a) zero (b) one (c) two (d) infinite

Options

A. zero
B. one
C. two
D. infinite

Answer

C

Explanation

`(c )` `A=({:(a,b),(c,d):})`
`A+A^(T)=I`
`implies({:(a,b),(c,d):})+({:(a,c),(b,d):})=({:(1,0),(0,1):})`
`implies2a=1`, `b+c=0`, `2d=1`
`impliesa=(1)/(2)`, `c=-b`, `d=(1)/(2)`
`impliesA=({:((1)/(2),b),(-b,(1)/(2)):})`
Now `AA^(T)=I`
`implies({:((1)/(2),b),(-b,(1)/(2)):})({:((1)/(2),-b),(b,(1)/(2)):})=I`
`implies({:((1)/(4)+b^(2),0),(0,b^(2)+(1)/(4)):})=({:(1,0),(0,1):})`
`impliesb^(2)+(1)/(4)=1impliesb=+-(sqrt(3))/(2)`
`:.A=({:((1)/(2),(sqrt(3))/(2)),(-(sqrt(3))/(2),(1)/(2)):})and({:((1)/(2),-(sqrt(3))/(2)),((sqrt(3))/(2),(1)/(2)):})`
No. of matrices `=2`