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1 1 The question is: Let $A = begin{bmatrix} A_1 \ A_2end{bmatrix}in mathbb{M}_{ntimes n}(mathbb{C})$ (an $ntimes n$ matrix with entries on $mathbb{C}$ ) and suppose that $mathcal{N}(A_1)=mathcal{R}(A_2^top)$ ( $mathcal{N}$ being the null space, and $mathcal{R}$ the range). Prove that $A$ is invertible. My thought process was, to prove that $A$ is invertible, it seems reasonable that from what we're given I'm gonna try to prove that $n(A)=0$ (the dimension of the null space of $A$ is $0$ ). Well, we have that $dim(mathcal{N}(A))=dim(mathcal{N}(A_1)capmathcal{N}(A_2))$ , which is equal to $dim(mathcal{N}(A_1))+dim(mathcal{N}(A_2))-dim(mathcal{N}(A_1)+mathcal{N}(A_2))$ . By the hypothesis, $mathcal{N}(A_1)=mathcal{R}(A_2^top)$ , and $mathcal{R}(A_2^top)=mathcal{R}(A_2)$ , so we're left with $$