- #1

RJLiberator

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## Homework Statement

Find 2x2 matrices A and B, all of whose entries are \begin{align} &\geq 0 \end{align}, such that A^-1 and B^-1 exist, but (A+B)^-1 does not exist.

## Homework Equations

The insverse is defined as 1/determinat(matrix) * adj(matrix)

Otherwise shown as:

[itex]\frac{1}{ad-bc}\begin{bmatrix}

d & -b \\

-c & a

\end{bmatrix}[/itex]

## The Attempt at a Solution

[/B]

My idea was to write it all out in unknown variable form. But I came to a problem.

[itex]\frac{1}{(a_1+a_2)(d_1+d_2)-(b_1+b_2)(c_1+c_2)}\begin{bmatrix}

(d_1+d_2) & -(b_1+b_2) \\

-(c_1+c_2) & (a_1+a_2)

\end{bmatrix}[/itex]

I then noted that the matrix inverse does

**not**exist if:

[itex](a_1+a_2)(d_1+d_2)-(b_1+b_2)(c_1+c_2)=0[/itex]

I then realized an issue with my method:

- There are many possible situations that this occurs.
- How do I check that A^-1 and B^-1 exist in this scenario.

Any helpful words of advice here?