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Chapter 7: Q47E (page 384)

TRUE OR FALSE

Similar matrices have the same characteristic polynomials.

Short Answer

Expert verified

True, Similar matrices have the same characteristic polynomials.

Step by step solution

01

Required data

Here, we need to show that there exists an invertible matrix.

02

AStep 2: Explanation for similar matrices have the same characteristic polynomials

If A and B are similar matrices, then there exists an invertible matrix S such that S-1AS = B.

From this we can get,

S-1(A - 位 I)S = S-1A S - S-1 位 I = B - 位 I

鈬� det (B - 位 I ) = det (S-1(A - 位 I)S )

= det S-1det (A - 位 I)det S = det (B - 位 I )

鈬� fA( 位 ) =fB( 位 )

So, A and B have the same characteristic polynomial.

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