Let I = I2 be the identity matrix of order 2 and let f(x) = |A − xI|. (In the study of matrices, f(x) is the characteristic polynomial of A,

Question

Let I = I₂ be the identity matrix of order 2, and let f(x) = |A − xI|. (In the study of matrices, f(x) is the characteristic polynomial of A, and the zeros of f(x) are the characteristic values (eigenvalues) of A.)

A= [ 3 4

5 4]

(a) Find the polynomial f(x). 

f(x) = 

(b) Find the zeros of f(x). (Enter your answers as a comma-separated list.) 

Answer 1

To solve |A - Ix|

first you must know the identity matrix.for a 2 by 2

It is alwaysa matrix with 1s on the main diagonal like so...

[1 0

0 1]

When you multiply this by x it changes to

[x 0

0 x]

Now subtract this from A

[3-x 4

5 4-x] (Yes it is that simple)

The determinant of this is

(3-x)*(4-x) - 4*5

The zeros of the above equation are when it = 0

(3-x)*(4-x) - 4*5=0

Solve this it is standard algebra.