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Goshia [24]
3 years ago
12

The trace , written tr(A) of an nxn matrix A is the sum of

Mathematics
1 answer:
Rus_ich [418]3 years ago
4 0

Answer:

Step-by-step explanation:

If A and B are nxn matrices then tr(AB)=tr(BA)... Hypothesis (1)

Prove:

A is similar to B

⇒There exists an invertible n-by-n matrix P such that B=P^{-1}AP.

⇒ tr[B]=tr[P^{-1}AP]

⇒tr[B]=tr[(P^{-1}A)P] Matrix multiplication has the associative property

⇒tr[B]=tr[P(P^{-1}A)]  Using hypothesis (1)

⇒tr[B]=tr[(PP^{-1})A] Matrix multiplication has the associative property

⇒tr[B]=tr[IA]   I is the identity matrix

⇒tr[B]=tr[A]      

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