Does there exist a 3 times 3 matrix A whose image is equal t

Does there exist a 3 times 3 matrix. A whose image is equal to its kernel as a subspace of R^3?

Solution

Let A be any 3x3 matrix.

Lwt us suppose that it is possible, image of A is equal to its kernel as subspace of R³.

Then,

dimension of image = dimension of kernel (1)

Since, we know that

Dimension of image + dimension of kernel = number of columns of A

using (1), we get

dimension of image + dimension of image = 3

2 dimension of image = 3

dimension of image = 3/2 = 1.5

which is not possible.

Hence, there does not exist such matrix whose image is equal to its kernel.