## Conditions on Coefficients that a Matrix is Nonsingular

## Problem 72

**(a)** Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that the entries of the matrix $A$ satisfy the following relation.

\[|a_{ii}|>|a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|\]
for all $1 \leq i \leq n$.

Show that the matrix $A$ is nonsingular.

**(b) **Let $B=(b_{ij})$ be an $n \times n$ matrix whose entries satisfy the relation

\[ |b_{i\,i}|=1 \hspace{0.5cm} \text{ and }\hspace{0.5cm} |b_{ij}|<\frac{1}{n-1}\]
for all $i$ and $j$ with $i \neq j$.

Prove that the matrix $B$ is nonsingular.

**(c)**

Determine whether the following matrix is nonsingular or not.

\[C=\begin{bmatrix}

\pi & e & e^2/2\pi^2 \\[5 pt]
e^2/2\pi^2 &\pi &e \\[5pt]
e & e^2/2\pi^2 & \pi

\end{bmatrix},\]
where $\pi=3.14159\dots$, and $e=2.71828\dots$ is Euler’s number (or Napier’s constant).

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