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in Matrices & determinants by (41.3k points)

Show that if A and B are symmetric and commute, then

(a) A-1B

(b) AB-1

(c) A-1B-1are symmetric.

1 Answer

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Best answer

(a) Since A and B commute: AB = BA

Pre- and post-multiplying both sides by A-1, we get

A-1(AB)A-1 = A-1(BA)A-1

⇒ (A-1A)(BA-1) = A-1B(AA-1) (by associativity)

⇒ I(BA-1) = (A-1B)I

⇒ BA-1 = A-1B

Now, (A-1B)′ = (BA-1′ = (A-1)′B′ (by reversal law)

= A-1B [as B′ = B (symmetric) and (A-1)′ = (A′) -1 = A-1]

Hence, A-1B is symmetric.

(b) Pre-and post-multiplying by B-1, we get

B-1(AB)B-1 = B-1(BA)B-1

⇒ (B-1A)BB-1 = B-1B(AB-1)

⇒ B-1A = AB-1

Now, (AB-1)′= (B-1A)′ = (A′B-1)′

= AB-1 [as A = A′ (symmetric) and (B-1)′ = (B′)-1 = B-1]

Hence, AB-1 is symmetric.

(c) Since A and B are symmetric, we have

AB = BA ⇒ (BA) -1 = (AB) -1

⇒ A-1B-1 = B-1A-1

⇒ (A-1B-1)′ = (B-1A-1)′ = (A-1)′ ⋅ (B-1)′ = A-1B-1

[as (A-1)′ = A-1 and (B-1)′ = B-1]

Hence, A-1B-1 is symmetric.

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