If and then prove that A² – A + 2I = 0.

Given: and

To prove A² – A + 2I = 0

Now, we will find the matrix for A², we get

[as c_ij = a_i1b_1j + a_i2b_2j + … + a_inb_nj]

Now, we will find the matrix for 2I, we get

So,

Substitute corresponding values from eqn(i) and eqn(ii), we get

[as r_ij = a_ij + b_ij + c_ij]

Therefore,

Hence proved