If then show that A is a root of the polynomial f(x) = x³ – 6x² + 7x + 2.

Given: and f(x) = x³ – 6x² + 7x + 2

To find the value of f(A)

We will substitute x = A in the given equation we get

f(A) = A³ – 6A² + 7A + 2I……………..(i)

Here I is identity matrix

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 A³, we get

So, Substitute corresponding values from eqn(i) and (ii) in equation f(A) = A³ – 6A² + 7A + 2I, we get

[as r_ij = a_ij + b_ij + c_ij],

Hence the A is the root of the given polynomial.