Show that the matrix is a root of the equation A² – 12A – I = 0.

Given:

I is an identity matrix 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 12A, we get

So,

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

[as r_ij = a_ij + b_ij + c_ij]

Therefore,

Hence matrix A is the root of the given equation.