What is the meaning of subtracting from the identity matrix? If I subtract the matrix $A$ from the identity matrix $I$, $I - A$, is there a meaning to the resulting matrix perhaps given some conditions like invertibility or symmetry? For example,
$$
\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
- \begin{bmatrix} a & b \\ c & d \end{bmatrix}
= \begin{bmatrix} 1-a & -b \\ -c & 1-d \end{bmatrix}
$$
For future reference, is there a good reference to lookup answers to such questions when I can't derive the answer on my own?

If A is invertible the resulting matrix B produced when you subtract $I$ from A may or may not be invertible. Can you think of examples?  If A is symmetric we have A = $A^t$, so, we have I - A = $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
- \begin{bmatrix} a & b \\ b & d \end{bmatrix}
= \begin{bmatrix} 1-a & -b \\ -b & 1-d \end{bmatrix}$ is also symmetric. You see something similar to this when you are calculating the characteritic polynomial of a matrix to calculate eigenvectors and test for the diagbalizability of a matrix. but instead you are looking at Det ( A - cI)= 0, where c is in your field. 
