query on diagonalizability of matrix

Question

Here I think matrix A can be an identity matrix . But it's answer is that A is diagonalizable . It's a single correct question . How can it be diagonalizable only . Why can't A be an identity matrix?

Answer 1

It can be an identity matrix, but you can't conclude that it is an identity matrix. There are lots of other possibilities, e.g. a diagonal matrix with any combination of $0$'s and $1$'s on the diagonal.

Answer 2

A matrix with this property is called a projection. Since the eigenvalues of $A$ satisfy $\lambda^2-\lambda=0$ their value is $0$ or $1$. Let $B = A - I$ (with I the identity matrix) then $B$ also is a projection, moreover $AB=BA=0$ so that A and B define projections on complementary subspaces. If you put A in its Jordan normal form then there are no Jordan blocks since they satisfy either $(x-I)^k = 0$ or $x^k=0$, so A is diagonalizable.