I have a symmetric positive semidefinite matrix $A$ that is irreducible, and with algebraic and geometric multiplicity for $\lambda(A) = 0$ as $1$. Another matrix $B$ with one entry in a diagonal is positive and all other zero, i.e., $B = {\rm diag}([0\, 0\, 0\, 1])$. Show that $A+B$ is a positive definite matrix.
In other words, will positive semidefinite matrix and irreducible $A$ become positive definite if a positive real value is added to one of its diagonal entries?
Let $$A=\begin{pmatrix}2 &1&1\\1&1&1\\ 1&1&1\end{pmatrix} \quad B= \begin{pmatrix}1 &0&0\\0&0&0\\ 0&0&0\end{pmatrix} $$ Then $\ker A$ is one dimensional and $A+B$ is not positive definite, as its determinant vanishes.