The Inverse Matrix of a Symmetric Matrix whose Diagonal Entries are All Positive
Problem 599
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.
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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.
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i feel that this solution is not rigorous enough because you are letting A be a specific matrix, so the result may not apply to all matrix cases.
Dear Lyqht,
I used a specific problem to show that the statement is FALSE. The statement is not true for all matrices. I proved this by giving a counterexample.
If you want to show that something is true for all matrices, then yes, we cannot use a specific matrix.
I hope this makes sense.