Let T be a linear operator on a finite-dimensional vector space, and let p(t) be the minimal polynomial of T. Prove the following results. (a) T is invertible if and only if p(0) - 0. (b) If T is invertible and p(t) = tn + a n _in _ 1 + + ait + a 0 , then T - 1 = - (T"- 1 + a n _,Tn - 2 + + a 2T + a,I)

MATH121 Chapter 2 Notes LESSON 2.1 – Linear Equations in One Variable Example 1. 6(5x - 5) = -31(3 - x) (Multiply 6 and 5x, and 6 and -5) (Multiply -31 and 3, and -31 and –x) 30x - 30 = -93 + 31x (Get similar values on same sides) -x = -63 (Divide by -1 to get x by itself) x = 63 (One Solution!) Example 2. 3.4x + 5 = 4.4x (Subtract 3.4x from 4.4x to get the similar values on the same side) 5 = x (One solution!) Example 3. (2z - 9)/5 – (1/10) = (-13z + 12)/10 (Since each is a fraction, you can easily see the similar denomina