Finding the Inverse of a Sum of Matrices

[CentreShifter]

Inverse of a sum of matrices [solved]

The problem is relatively simple. Given the equation:

$$(I+2A)^{-1}= \begin{bmatrix} -1 & 2 \\ 4 & 5 \end{bmatrix}$$

Find A.

My problem seems to be that I'm distributing the inverse on the LHS incorrectly. My real question then is, is the following correct?

$$(I+2A)^{-1}=I^{-1}+\frac{1}{2}A^{-1}$$

 

[Mark44]

The problem is relatively simple. Given the equation:

$$(I+2A)^{-1}= \begin{bmatrix} -1 & 2 \\ 4 & 5 \end{bmatrix}$$

Find A.

My problem seems to be that I'm distributing the inverse on the LHS incorrectly. My real question then is, is the following correct?

$$(I+2A)^{-1}=I^{-1}+\frac{1}{2}A^{-1}$$

No. There is no distributive property for exponents, which is what you seem to be doing.

Since
$$(I+2A)^{-1}= \begin{bmatrix} -1 & 2 \\ 4 & 5 \end{bmatrix}$$

The matrix on the right is clearly invertible, so why don't you take the inverse of both sides?

 

[Hurkyl]

Well, exponents do distribute over products in some fashion, and there is the binomial theorem and its generalizations. But the first is not relevant and the second a long and torturous path for this problem.

 

[Mark44]

I should have been more clear - that exponents don't distribute over a sum; in other words, that (A + B)ⁿ $\neq$ Aⁿ + Bⁿ.

 

[CentreShifter]

Beautiful.

Inverting both sides did the trick.

Inverse of the RHS is $$\begin{bmatrix}\frac{-5}{13} & \frac{2}{13} \\ \frac{4}{13} & \frac{1}{13} \end{bmatrix}$$. I'll call this $$B^{-1}$$

So then I'm left with
$$\begin{align}I+2A&=B^{-1} \\ 2A&=B^{-1}-I \\ A&=\frac{B^{-1}-I}{2}=\begin{bmatrix} \frac{-9}{13} & \frac{1}{13} \\ \frac{2}{13} & \frac{-6}{13}\end{bmatrix} \end{align}$$

Plugging this into the original LHS yields the correct result. Thanks.