Apologies for the low quality of the attached image.
Presented is an attempt to calculate the product of six matrices of size 3x3. The product: {{1,0,0},{0,1,0},{0,0,9}}
I then split the multiplication into two halves: The product of the first half: {{1,0,0},{-1,1,0},{2,-4,1}} The product of the second half: {{1,4,0},{0,1,3},{0,0,9}}
Multiplying these two matrices yields {{1,4,0},{-1,-3,3},{2,4,-3}}, which is correct. This is in fact the matrix which I row reduced to obtain the six elementary matrices in the original product.
Is there a reason why Wolfram|Alpha is computing the first result?
Six matrices
First three
Second three
First three, second three combined
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