If `A` is a matrix such that `A^2+A+2I=Odot,` the which of the following is/are true? (a) A is non-singular (b) A is symmetric (c) A cannot be skew-symmetric (d) `A^(-1)=-1/2(A+I)`

Options

A. A is nonsingular
B. A is symmetric
C. A cannot be skew-symmetric
D. `A^(-1)=-1/2 (A+I)`

Explanation

Given, `A^(2)+A+2I=O`
`implies A^(2)+A=-2I`
`implies |A^(2)+A|=|-2I|`
`implies |A||A+I|=(-2)^(n)`
`implies |A| ne 0`
Therefore, A is nonsingular, hence its inverse exists. Also, multiplying the given equation both sides with `A^(-1)`, we get
`A^(-1) =-1/2 (A+I)`