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

##### 2 Answers

See below.

#### Explanation:

Write it as:

Where

And then process the algebra.

If you need an answer check, I get:

#### 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))#

**Footnote**

Note that

#x^2-7 = 0#

As a result, we find that the set of matrices of the form *field* under matrix addition and multiplication. This field is essentially the same as (i.e. *isomorphic to*) the set of numbers of the form

Note that this matrix