Matrix question

Question

Let

     (    3      1      -2  )

A= (   -2     2       2  )

      (   0      1       1  )

A is a 3x3 matrix. Find all the values of y such that det(A - yI)=0, where I is the 3x3 identity  matrix.

Thank you

Answer 1

A-yI=

( 3-y  1   -2 )

(  -2 2-y   2 )

( 0    1  1-y )

The determinant evaluates as (3-y)((2-y)(1-y)-2)+2(1-y)-2(-2)=

(3-y)(2-3y+y²-2)+2-2y+4=0.

(3-y)(-3y+y²)+2(3-y)=0.

(3-y)(y²-3y+2)=(3-y)(y-1)(y-2)=0. Therefore, y=1, 2 or 3.