Let $A$ =$\left( \begin{array}{cc} a & b \\ c & d \end{array} \right) $ with $a,b,c,d$ positive real numbers such that $a+b<1/2$ and $c+d<1/2$

Question

Let $A$ =$\left( \begin{array}{cc} a & b \\ c & d \end{array} \right) $ with $a,b,c,d$ positive real numbers such that $a+b<1/2$ and $c+d<1/2$, then which of the following statements is FALSE?

A. $I+A$ invertible

B. $I-A$ invertible

C. $I+A^2$ invertible

D.there exists a natural number $m$ such that $I+A^m$ is not invertivle.

What I did I set some values of $a,b,c,d$ to test that option A and B FALSE. I am stuck in C and D option. Please help.

Answer 1

What I did I set some values of $a,b,c,d$ to test that option A and B FALSE.

This doesn't sound right. By Gerschgorin's disc theorem, the eigenvalues of $A$ lies inside the union of closed discs $D(a,b)\cup D(d,c)$. Since $a,b,c,d>0$, the maximum distance from $D(a,b)\cup D(d,c)$ to the origin is $\max(a+b,c+d)$, which is smaller than $\frac12$. Hence $|\lambda|<\frac12<1$ for every eigenvalue $\lambda$ of $A$. It follows that the statements in A,B,C are true and the statement in D is false.