Calculating Trace of 4x4 Matrix: (A+I)^{-1}

[Ahmes]

Hello,
I have to calculate the trace of the following matrix: (A+I)^{-1}
where A is:
1 0 0 1
0 2 2 0
0 2 2 0
1 0 0 1
and I is the unit matrix 4x4.
The calculations are extremely excruciating and the result is 38/15 [checked]. I'm afraid I miss the whole point of this. Can the fact the matrix is diagonalizable help me reach the answer faster?
I mean not going through the process of adding 1 to the diagonal elements and do the inverse.

Thank you.

 

[Hurkyl]

Can the fact the matrix is diagonalizable help me reach the answer faster?

It would if you could get the trace of A from its diagonalization!