Real n-by-n Matrices...

Suppose exists $B$ such that $AB=I$ and that $m$ is the smallest integer such that, exist coefficients $t_i$ such that, $$ t_0 I + t_1 A + \cdots + t_m A^m = 0 $$

If $t_0=0$ you have, $$ t_1 A + \cdots + t_m A^m = 0 $$

Now, see if you can use the fact that $AB = I$ to find new coefficients $t_i'$ satisfying $$ t_0'I + t_1'A + \cdots t_n ' A^n = 0 $$ where $n<m$.

Since by assumption $m$ is the smallest integer such that this type of combination exists, then you have a contradiction.

I liked that book as a first course in linear algebra. You could also see if past exam problems are available to study from. To me, studying for exams is somewhat dependent on the format of exam itself so its hard to give general advice.

Hint: assume $t_0=0$ then $$ t_1A+\ldots+t_mA^m=0. $$ Multiply by $B$ and get the contradiction with the minimality of $m$.