Prove trace of matrix: Tr(AB) = Tr(BA)

Homework Statement

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The trace of a matrix is defined to be the sum of its diaganol matrix elements.
1. Show that Tr(ΩΛ) = Tr(ΩΛ)
2. Show that Tr(ΩΛθ) = Tr(θΩΛ) = Tr(ΛθΩ) (the permutations are cyclic)

my note: the cross here U[+][/+]is supposed to signify the adjoint of the unitary matrix U

Homework Equations

$$
Tr(Ω) = \sum_{i=1} Ω_{ii} \\
I = \sum_{k=1}^n | k >< k |
$$

The Attempt at a Solution

$$
Tr(Ω) = \sum_{i=1} Ω_{ii} \\
Tr(ΛΩ) = \sum_{i=1} (ΛΩ)_{ii} \\
(Ω)_{ij} = < i | Ω | j >
$$

(this is saying that when we take the product of the matrices we sum the diagonal entries where the element is in the ith row of the ith column, I also assume the trace is )

1.
$$
= \sum_{i=1} (Λ Ω)_{ii} \\
= \sum_{i=1} < i |ΛΩ| i > \\
= \sum_{i=1} < i |Λ I Ω| i > \\
= \sum_{k=1} \sum_{i=1} < i |Λ| k >< k |Ω| i > \\
= \sum_{k=1} \sum_{i=1} Λ_{ik} Ω_{ki}
$$

Unsure about how to finish as I think I am on the right track but my thinking is a bit cloudy.

To finish proof i need to show as this is where we end up if we reverse the operators initially as we want:

$$
\sum _{k=1} \sum _{i=1} Λ_{ik} Ω_{ki} = \sum _{k=1} \sum _{i=1} Ω_{ki} Λ_{ik}
$$

So correct me where/if I’m wrong.

The way I’m thinking about it is in terms of matrix multiplication. The trace only sums the diagonal elements of the a matrix. Thus when multiplying two matrices/operators the only terms that ‘survive’ are terms which end up there as the ith row * the ith column. This is denoted by the fact that ‘ i ‘ is the 1st term $$Λ_{ik}$$ denotes the ith row time of Λ times the ith column $$Ω_{ki}$$ when we sum through all the values k = 1 to k = n.

Is it possible that we can just swap the two indexes k & i? as they both go to n? Can I just swap their positions as they are are matrix elements and are thus commutative?

For part 2 i can use a similar method to get to:
$$
\sum _{k=1} \sum _{i=1} \sum _{j=1} Λ_{ik} Ω_{kj} θ_{ji}
$$
I notice that the last index of a matrix matches the first index of the next operator. Which may be a clue to why its cyclic but I can’t figure out why.

Apologies for the long winded attempt, I just wanted to be clear for anybody attempting to understand my confusion. In my text this proof is in the Unitary Matrix/ Determinant section. These are obviously relevant in later parts to this question but they don’t appear to be relevant here.

Any help greatly appreciated