- #1
RJLiberator
Gold Member
- 1,095
- 63
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?