If show that A² – 5A + 7I₂ = 0.

Given:

I₂ is an identity matrix of size 2, so

To show that

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 5A, we get

Now,

So,

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

[as r_ij = a_ij + b_ij + c_ij]

Therefore,

Hence proved