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Neither did our professor nor book give a proof of this fact, but just in case I need to know that.

amWhy
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A.boj
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    When neither your professor nor your book prove it, that may be a sign to look elsewhere. For example Wikipedia (https://en.wikipedia.org/wiki/Determinant#Multiplicativity_and_matrix_groups). Or it may be a sign that the proof will occur later. – GEdgar Sep 07 '18 at 00:08
  • While I suspect this question is somewhat too broad and unmotivated, I'll give you a big hint: what does the derivative of a linear transformation represent? – Steven Stadnicki Sep 07 '18 at 00:08
  • (That's not to say that you're unmotivated! Just that the question comes out of nowhere with no background information.) – Steven Stadnicki Sep 07 '18 at 00:09
  • @StevenStadnicki I don’t see the point of this question – Thinking Sep 07 '18 at 00:13
  • This is true for all square matrices, not all matrices. – MasB Sep 07 '18 at 00:29
  • @Thinking because with the 'geometric' definition of the derivative of a transform, along with the understanding that $AB$ is the transform that corresponds to performing transform $B$, then performing transform $A$ on the result, the result becomes (IMHO) easy to understand. – Steven Stadnicki Sep 07 '18 at 05:09

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First you prove this for all $B$ and all elementary matrices $A$ (an elementary matrix is a matrix that is obtained from $I$ by using one elementary row operation).

Next, you show that multiplying by $A$ is the same as applying said elementary row operation.

After that, you study the effect of each elementary row operation on the determinant. Then you have proved the theorem for: arbitrary $B$, and elementary matrix $A$.

After that, you show that every invertible matrix is a product of elementary matrices. This ensures that the theorem is true when $A$ is invertible.

When $A$ is not invertible, treat that case separately. (You have to show that the determinant is zero, once again, by row-reducing.)

amWhy
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Mark
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The result known as Cauchy-Binet Formula and holds in a more general context for $A$ m-by-n and $B$ n-by-m matrices and of course also in the special case of $m=n$ for square matrices.

user
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