Let $A$ be a real symmetric matrix whose diagonal entries are all positive real numbers.

Is it true that the all of the diagonal entries of the inverse matrix $A^{-1}$ are also positive?
If so, prove it. Otherwise, give a counterexample.

Solution.

The statement is in general false. We give a counterexample.

Let us consider the following $2\times 2$ matrix:
\[A=\begin{bmatrix}
1 & 2\\
2& 1
\end{bmatrix}.\] The matrix $A$ satisfies the required conditions, that is, $A$ is symmetric and its diagonal entries are positive.

The determinant $\det(A)=(1)(1)-(2)(2)=-3$ and the inverse of $A$ is given by
\[A^{-1}=\frac{1}{-3}\begin{bmatrix}
1 & -2\\
-2& 1
\end{bmatrix}=\begin{bmatrix}
-1/3 & 2/3\\
2/3& -1/3
\end{bmatrix}\] by the formula for the inverse matrix for $2\times 2$ matrices.

This shows that the diagonal entries of the inverse matrix $A^{-1}$ are negative.

The post The Inverse Matrix of a Symmetric Matrix whose Diagonal Entries are All Positive first appeared on Problems in Mathematics.