# 2015-17 Inverse of a minor

Let $$H$$ be an $$N \times N$$ positive definite matrix and $$G = H^{-1}$$. Let $$H’$$ be an $$(N-1) \times (N-1)$$ matrix obtained by removing the $$N$$-th row and the column of $$H$$, i.e., $$H’_{ij} = H_{ij}$$ for any $$i, j = 1, 2, \cdots, N-1$$. Let $$G’ = (H’)^{-1}$$. Prove that
$G_{ij} – G’_{ij} = \frac{G_{iN} G_{Nj}}{G_{NN}}$
for any $$i, j = 1, 2, \cdots, N-1$$.

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# 2013-21 Unique inverse

Let $$f(z) = z + e^{-z}$$. Prove that, for any real number $$\lambda > 1$$, there exists a unique $$w \in H = \{ z \in \mathbb{C} : \text{Re } z > 0 \}$$ such that $$f(w) = \lambda$$.

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Let $$A=(a_{ij})$$ be an $$n\times n$$ upper triangular matrix such that $a_{ij}=\binom{n-i+1}{j-i}$ for all $$i\le j$$. Find the inverse matrix of $$A$$.