In a problem in my textbook I'm asked to find the inverse and the determinant of the following matrix:
$\begin{bmatrix}1 & 0 & 0 & 0 & 0\\0 & 1& 0& 0& 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & a & b\\ 0 & 0 & 0 & c & d\end{bmatrix}$
The determinant should be $ad-bc$, that part is simple. As for the inverse, my intuition is to create an augmented matrix (with the identity matrix to the right) and focus on the two bottom rows (because naturally, the inverse of a diagonal of ones surrounded by zeros would just be another diagonal of ones surrounded by zeros). The inverse of the variables to the bottom right ought to be some pretty hairy algebraic expression with $ad-bc$ in the denominator.
Much to my surprise, the key states that the inverse of the matrix is:
$\begin{bmatrix} ad-bc & 0 & 0 & 0 & 0\\ 0 & ad-bc& 0& 0& 0\\ 0 & 0 & ad-bc & 0 & 0\\ 0 & 0 & 0 & d & -b\\ 0 & 0 & 0 & -c & a\end{bmatrix}$
At least to my eyes, this seems to be patently false. The definition of an inverse is that $AA^{-1}=I$. I need only look at the first column of the composite matrix $AA^{-1}$ to verify that this can't be true. The first element in the column is
$$1*(ad-bc) +0*0+0*0+0*0+0*0 =ad-bc\neq 1$$
....Which means we're clearly not dealing with an identity matrix, unless of course $ad-bd=1$, but that isn't stated in the text.
Is this a misprint or am I missing something myself?