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If I have two matrices, one of which is $3 \times 2$ and the second is $2 \times 3$. How can I find the determinant of the square matrix that results from multiplying them without multiplying them and getting the resulted matrix in order to get the determinant of it?

Sammy Black
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Lduh
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  • The resulting matrix will be $3 \times 3$, what do you know about the rank of a product in terms of the ranks of the individual terms? – Padraig Ó Catháin Mar 11 '24 at 17:10
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    Hint: What's the maximum rank of a $3 \times 2$ or a $2 \times 3$ matrix? – Sammy Black Mar 11 '24 at 17:10
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    @Lduh Note that askers are expected to provide context for their questions, as is explained here. Please [edit] your question to add more information such as where you encountered this question and what you've tried so far. If you don't make this change, it is likely that your question will be closed. – Ben Grossmann Mar 11 '24 at 17:14
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    Incidentally, for a product of a $2 \times 3$ matrix with a $3 \times 2$ matrix, one could use the Cauchy-Binet inequality. – Ben Grossmann Mar 11 '24 at 17:22
  • It’s my first time here sorry I didnt put the question properly – Lduh Mar 11 '24 at 17:26
  • You may want to check Cauchy–Binet formula. – Salcio Mar 11 '24 at 17:47
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    Highly related: https://math.stackexchange.com/questions/2620133 – Anne Bauval Mar 11 '24 at 18:11
  • Think of determinant as a ratio of volumes, the first mapping by the $3\times 2$ maps a volume onto an area and the $2 \times 3$ will map that area to an area embedded in 3-D – WW1 Mar 11 '24 at 19:56

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