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Reckoning of Sand

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## Homework Statement

Let [itex]A[/itex] be an [itex]nxn[/itex] matrix, and [itex]C[/itex] be an [itex]mxm[/itex] matrix, and suppose [itex]AB = BC[/itex].

(a) Prove the following by induction: For every [itex]n∈ℕ[/itex], [itex](A^n)B = B(C^n)[/itex]. What property of matrix multiplication do you need to prove this?

## Homework Equations

The four basic properties of matrix multiplication discussed in my course are

1. Distributive: [itex](A + B)C = AC + BC[/itex] and [itex]C(A + B) = CA + CB[/itex]

2. Scalar Commutativity: [itex](tA)B = t(AB) = A(tB)[/itex]

3. Associative: [itex]A(BC) = (AB)C[/itex]

## The Attempt at a Solution

If [itex]n = 1[/itex], then [itex](A^n)B = B(C^n)[/itex] becomes [itex](A^1)B = B(C^1)[/itex] which is[itex] AB = BC[/itex] which is supposed to be true.

Assume [itex](A^n)B = B(C^n)[/itex] is true for [itex]n = k[/itex]. Then [itex](A^k)B = B(C^k)[/itex] is true.

Prove [itex](A^n)B = B(C^n)[/itex] is true for [itex]n = k + 1[/itex].

[itex](A[/itex]

^{k+1}[itex])B = B(C[/itex]

^{k+1}[itex]) ⇒ (A(A^k))B = B(C(C^k))[/itex]

I'm not really sure how to proceed. I tried rewriting the matrix products as the summations of the products of their entries, but that didn't seem to get me anywhere. I know that I have to relate the equation when [itex]n = k + 1[/itex] to the equation when [itex]n = k[/itex] as part of induction, but I'm still stuck.