Hi, is there a reason that you aren't replacing both P matrices with P * transpose(P), or is it just implied that we're doing that to both sides? Thanks.
Added by Trinidad H.
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Key Concepts
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Let $A$ and $P$ be $n \times n$ matrices, where $P$ is invertible. Does $P^{-1} A P=A ?$ IIlustrate your conclusion with appropriate examples. What can you say about the two determinants $\left|P^{-1} A P\right|$ and $|A| ?$
Determinants
Properties of Determinants
What is wrong with this proof that projection matrices have det $P=1$ ? $$ P=A\left(A^{\mathrm{T}} A\right)^{-1} A^{\mathrm{T}} \quad \text { so } \quad|P|=|A| \frac{1}{\left|A^{\mathrm{T}}\right||A|}\left|A^{\mathrm{T}}\right|=1 $$
Properties of the Determinant
In Exercises 31–36, mention an appropriate theorem in your explanation. Let $A$ and $P$ be square matrices, with $P$ invertible. Show that $\operatorname{det}\left(P A P^{-1}\right)=\operatorname{det} A$
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