Hello I would greatly appreciate if anyone could help me with the answers to que

Question

Hello I would greatly appreciate if anyone could help me with the answers to question 6a-6L(6a,6b,6c,6d,6e,6f,6g,6h,6i,6j,6k,6l). I really need help answering these listed question not number 4 or 5 please help. Note 6i-6L is listed on the top of the 2nd page/image below the 1st page/image. Thank you for your efforts.

The facts in are shown product to define algebraic and t the Jordan product commutative make for structures cal is Commutator g results in this area of The proverio of Product Nordan Definition: The commutator product. or matrix cross product of two n by n matrices and B is Questions: 4. Why must both A and B be square matrices? 5. culate the cross product of the following pairs of matrices. and d) 3 and 6 6. Either prove that the following true a counterexample to show that the properties are or give property does not always All matrices below are assumed to be n n, I is the n n hold. identity matrix. O n x n zero matrix, any scalar and r is b) A x B -(B x A) e) (A x B) x C A x (B x C) g) (A B) x C (A x C) (B x C) h) A x (B+ C) A x B) t (A x C)

Explanation / Answer

4) Why must both A and B be square matrices?

The commutator of two matrices, A and B, denoted by [A,B] is the matrix C=AB-BA.


If A is r x k, then B needs to be k x m in order to multiply AB. But we also need to be able to find BA.. B is k x m so A must have m rows. That means that m=r.

So A is r x k and B is k x r, then AB is r x r and BA is k x k.
However, the commutator says we need to be able to multiply and then SUBTRACT. To subtract matrices, they have to be the same size, so r=k.

Thus, A and B must both be square matrices of the same size.

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