If C is 3 x 2 matrix, D is 2 x 3 matrix, and E is 3 x 3 matrix, that K should be 2 x 2 matrix if CKD = E. With specific entries of each matrices, I could get the elements of K with expanding the equation with matrix multiplication. But, is there another way to determine K that briefer than to expanding that equation? Because matrix C and D can't be inverted due to the different numbers of each rows and columns.
Delete the third rows of C and E, and the third columns of D and E.
Perhaps a different row and column instead if necessary.
Multiply both sides by $C'$ (from the left), where the first row of $C'$ is orthogonal to the second column of $C$, and the second row of $C'$ is orthogonal to the first column of $C$. Such vectors exist and are easy to find.
Multiply both sides (from the right) by $D'$ whose first column is orthogonal to the second row of $D$, and vice versa.
We get $C'CKDD'=C'ED',$ where $C'C$ and $DD'$ are $2×2$ diagonal matrices. It is quick to finish.