Hello. How can I find an example of a matrices $A,B,C,D$ so that the determinant of the block matrix
$$\begin{vmatrix}A & B \\ C & D \end{vmatrix} \neq \begin{vmatrix} AD-CB \end{vmatrix}$$
This can happen if $AC \neq CA$ but I've been trying for an hour simple $2 \times 2$ matrices and every I tried so far with $AC \neq CA$ still satisfies the above determinant relation. How would you go about finding an example which doesn't work?
Consider the transforms
$$ A(f(x),g(x)) = h(x) = 1 + \int_0^x f(x - t) g(t) dt $$
$$ B(h(x),g(x)) = f(x) $$
$$ C(h(x),f(x)) = g(x) $$
Where the functions on the LHS are given.
Notice $B,C$ are the inverses of $A$.
How to compute $B,C$ ??
Is there an integral representation for $B$ or $C$ ?
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