If A is any square matrix such that `A+I/2` and `A-I/2` are orthogonal matrices, then

Options

A. A is orthogonal
B. A is skew- symmetric matrix of even order
C. `A^(2)= 3/4 I`
D. None of these

Answer

B

Explanation

`because (A-1/2I)(A-1/2)^(T)=I ` ...(i)
and `because (A+1/2I)(A+1/2)^(T)=I ` ...(ii)
`rArr (A-1/2I)(A^(T)-1/2)=I `
and `rArr (A+1/2I)(A^(T)+1/2)=I `
`rArr A + A^(T) = 0 ` [subtracting the two results]
` rArr A^(T) = - A`
`therefore` A is skew-symmetric matrix.
From first result, we get
`A A ^(T) = 3/4 I`
`rArr A^(2) = - 3/4 I`
`therefore abs(A^(2) ) = abs(-3/4I)`
`therefore abs(A)^(2) = (-3/4)^(n)`
`rArr n` is even.