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5 Let $M_n(mathbb{R})$ denote the vector space of real $ntimes n$ matrics, and let $A in M_n(mathbb{R})$ . Part (a) of this question says: Suppose $B in M_n(mathbb{R})$ such that $AB = I_n$ (the $n times n$ identity matrix. If $C in M_n(mathbb{R})$ such that $CA = 0$ , then prove $C = 0$ . I have already proven part (a). Part (b) asks: Assume there exists a least positive integer $m$ such that $t_0I + t_1A + dots + t_mA^m = 0$ for some $t_0, dots, t_m in mathbb{R}$ with $t_m neq 0$ . Also, suppose that $AB = I_n$ for some $B in M_n(mathbb{R})$ .Prove that $t_0 neq 0$ . (Hint: Use the result from part (a)). My idea is to use induction on $m$ . That is, suppose $$t_0I = 0.$$ But this implies that $t_0 = 0$ since $I$ is the identity. But we could see this as begin{align*} t_0I &= 0 \ t_0AB &