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For a matrix to be positive definite (PD), all its leading principal minors must be positive. If they are all non-negative (some may be zero), the matrix is positive semi-definite (PSD). --- Matrix A: $$ A = \begin{bmatrix}1 & 1 & 0\\ 1 & 2 & 1\\ 0 & 1 & 1\end{bmatrix} $$ 1. First leading principal minor: $\Delta_1 = 1 > 0$ ✅ 2. Second leading principal minor: $$ \Delta_2 = \begin{vmatrix}1 & 1 \\ 1 & 2\end{vmatrix} = (1)(2) - (1)(1) = 1 > 0 \] ✅ 3. Third leading principal minor (determinant of A): \[ \det(A) = 1 \begin{vmatrix}2 & 1\\ 1 _______ _____ __________ ______ _____ ____ ____ _________ ____ _________ _________.
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