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Minchanka [31]
4 years ago
10

Show that if A and B are similar nxn matrices, then det(A)=det(B).

Mathematics
1 answer:
natita [175]4 years ago
7 0

Step-by-step explanation:

To prove it we just use the definition of similar matrices and properties of determinants:

If A,B are similar matrices, then there is an invertible matrix C, such that A=C^{-1}BC} (that's the definition of matrices being similar). And so we compute the determinant of such matrix to get:

det(A)=det(C^{-1}BC)=det(C^{-1})det(B)det(C)

=\frac{1}{det(C)}det(B)det(C)=det(B)

(Determinant of a product of matrices is the product of their determinants, and the determinant of C^{-1} is just \frac{1}{det(C)})

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