An Insignificant question: Find non-zero operator K so det[K(A+B)] = det[K(A)] + det[K(B)], A,B are 3-by-3 matrices

[NotWindowsExplorer]

Find a non-zero operator [imath]K[/imath] that acts on a 3x3 matrix so that
[math]\mathrm{det}[K(A+B)]=\mathrm{det}[K(A)]+\mathrm{det}[K(B)][/math]where [imath]A[/imath] and [imath]B[/imath] are any 3x3 matrix.

You can also put down any operators you found that partially satisfy this property, i.e. [imath]A[/imath] & [imath]B[/imath] needs to be invertible, or [imath]A[/imath] & [imath]B[/imath] needs to have the same determinant value

 

[blamocur]

What kind of operator can [imath]K[/imath] be? Multiplication by a matrix ? Arbitrary linear transform? Something else?

 

[NotWindowsExplorer]

What kind of operator can [imath]K[/imath] be? Multiplication by a matrix ? Arbitrary linear transform? Something else?

[imath]K(M)[/imath] which represents operator [imath]K[/imath] acting on matix [imath]M[/imath] could be like a power series like
[math]c_0 \times I+c_1 \times M+c_2 \times M^2+c_3 \times M^3+c_4 \times M^4+...[/math]where [imath]I[/imath] is the identity matrix of the same dimension or [imath]K[/imath] is an operator that simply multiplies [imath]M[/imath] by a matrix [imath]k[/imath].

 

[Otis]

where A and B are any 3x3 [matrices]

That seems like a bold claim.

put down any operators you found that partially satisfy [the equation], i.e. A & B [need] to be invertible, or…

Placing conditions on A and B seems more reasonable. I say that because I'd started by thinking about the simpler problem

det[A+B] = det[A] + det[B] …… where A and B are 2×2 matrices

and it didn't take long to realize that the equation is not generally true.

After picking symbols for the elements of A and B, I wrote out expressions for det[A]+det[B] and for det[A+B]. They showed that the equation is true only when four specific terms in those expressions sum to zero. It is clear enough how to make that happen (i.e., how to choose elements of B, given elements of A). However, the success of forging B like that depends on a prerequisite condition: none of the elements in A or B can be zero. (There could be other prerequisite conditions; I didn't write a proof.)

In other words, the equation det[A+B]=det[A]+det[B] is not true for any pair of 2×2 matrices. Finding a single matrix operator K to always make it true seems unlikely. By extension, the claim in red above seems like a stretch.

Was this exercise given to you in class?