Question

If `A=[(-1,2), (3,1)]`, how do you find F(A) where `f(x)=x^2-2x+3`?

Answer 1

`2 [(6 , -2),(-3 , 4)]`

See below.

Explanation:

Write it as:

`f(A)=[(-1,2), (3,1)]cdot[(-1,2), (3,1)] - 2[(-1,2), (3,1)] + 3 I`

Where `I` is the identity matrix: `[(1,0), (0,1)]`

And then process the algebra.

If you need an answer check, I get:

`f(A) = 2 [(6 , -2),(-3 , 4)]`

Answer 2

`f(A) = A^2-2A+3I = ((12,-4),(-6,8))`

Explanation:

Given:

`A = ((-1, 2),(3, 1))`

Then:

`A^2 = ((-1, 2),(3, 1))((-1, 2),(3, 1)) = ((7, 0),(0, 7)) = 7I`

So:

`A^2-2A+3I = 7I-2A+3I`

`color(white)(A^2-2A+3I) = 10I-2A`

`color(white)(A^2-2A+3I) = ((10,0),(0,10))-((-2,4),(6,2))`

`color(white)(A^2-2A+3I) = ((12,-4),(-6,8))`

`color(white)()`
Footnote

Note that `A` is a "square root of `7`". That is, it is a root of the polynomial:

`x^2-7 = 0`

As a result, we find that the set of matrices of the form `pI+qA` where `p` and `q` are rational is a field under matrix addition and multiplication. This field is essentially the same as (i.e. isomorphic to) the set of numbers of the form `p+qsqrt(7)` for rational multipliers `p, q`.

Note that this matrix `A` is not the only possible way to represent `sqrt(7)` with a rational matrix. For example, you could use `((0, 7),(1, 0))`.