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Lyrx [107]
3 years ago
6

Suppose that A = PDP-1 . Prove that det(A) = det(D)

Mathematics
1 answer:
daser333 [38]3 years ago
5 0

Answer:

Check.

Step-by-step explanation:

To prove it we need to know that  for two matrices A and B we have that:

det(AB) = det(A)*det(B) and det(A^{-1}) = \frac{1}{det(A)}. Now:

A = PDP^{-1}

det(A) = det(PDP^{-1})

det(A) = det(P)*det(D)*det(P^{-1})

det(A) = det(P)*det(D)*\frac{1}{det(P)}

det(A) = det(P)*\frac{1}{det(P)}*det(D)

det(A) = det(D).

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jeka57 [31]

Answer:

\\ \gamma= \frac{a\cdot b}{b\cdot b}

Step-by-step explanation:

The question to be solved is the following :

Suppose that a and b are any n-vectors. Show that we can always find a scalar γ so that (a − γb) ⊥ b, and that γ is unique if b \neq 0. Recall that given two vectors a,b  a⊥ b if and only if a\cdot b =0 where \cdot is the dot product defined in \mathbb{R}^n. Suposse that b\neq 0. We want to find γ such that (a-\gamma b)\cdot b=0. Given that the dot product can be distributed and that it is linear, the following equation is obtained

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By construction, this γ is unique if b\neq 0, since if there was a \gamma_2 such that (a-\gamma_2b)\cdot b = 0, then

\gamma_2 = \frac{a\cdot b}{b\cdot b}= \gamma

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Step-by-step explanation:

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