Let $A,B$ be matrices, their entries are real numbers. If $A,B$ are square matrices with the same orders then $$ \det\left(A^{T}BA\right)=\det\left(B\right)\left(\det\left(A\right)^{2}\right). $$ My question is, if their orders are not the same but $B$ is still square, do we have a formula in the form like this $$ \det\left(A^{T}BA\right)=\det\left(B\right)\times\ldots, $$ here $\ldots$ is something that is interseting enough.
A question for a matrix equality.
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linear-algebra
matrices
linear-transformations
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0For example, A is 3x2; B is 3x3. – 2017-02-07
2 Answers
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No,See that determinant is always defined for Square matrices only , in your example even though the final outcome for $A^{T}BA$ may be a square mtrix and of course we can calculate its determinant but we cannot write them as individual factors as $A$ is not a square matrix here.
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0I know we cannot have det(A) but may be something else. – 2017-02-07
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0Ok, since finally determinants are numbers you can decompose the determinant of $A^{T}BA$ into factors and you can write them as determinants of some other matrix [matching of corresponding factors] by trial method ! . – 2017-02-07
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No this formula works only for i × j where i = j.
And both A and B are of same order i × j.