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 cij = ai1b1j + ai2b2j + … + ainbnj]
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 rij = aij + bij + cij],
Hence the A is the root of the given polynomial.