For finding eigen vectors, we use the characteristic equation which is a homogeneous system of linear equations
[ A-kI ]X =0
where k is eigen value and I is identity9 matrix and X is eigen vector corresponding to eigen value k.
For given matrix which is a identity matrix of order 2, eigen values are 1,1 ie repeated igen values. Here k=1 only.
First find the matrix [ A-kI ] and find the rank of matrix. Let rank of matrix be r.
Here value of r is 0 as [A-kI] will be null matrix of order 2.
Then number of linearly independent eigen vectors= number of linearly independent solutions of homogeneous set of equations= n-r
where n is the order of given matrix
Here number of LI eigen vectors= 2-0=2