# 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.

## 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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### 2 Responses

1. Lyqht says:

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.

• Yu says:

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.

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