Which of the following matrices is/are diagonally dominant?
(a) \(\begin{bmatrix}-2 & 5 & 0\\ 0 & 1& 1\\ -2 & -1& 1\end{bmatrix}\qquad\) (b) \(\begin{bmatrix}-2 & 1 & 0\\ 0 & 1& 1\\ 1 & -1& 2\end{bmatrix}\)
(c) \(\begin{bmatrix}-2 & 5 & 8\\ 0 & 1& 1\\ 0 & -1& 1\end{bmatrix}\qquad\) (d) \(\begin{bmatrix}1 & 1 & 2\\ 0 & 1& 1\\ -2 & -1& 1\end{bmatrix}\)
Explanation: (b) is correct options. A square matrix \(A\) is called diagonally dominant if $$|a_{ii}|\geq \Sigma_{j=1 \atop i \ne j}^n|a_{ij}|, \text{ for all } i=1,...,n.$$
Tags:
Numerical