Question

if A&B are orthogonal,then (AB)transpose (AB) is equal to​

Answer 1

If A & B are orthogonal , then (AB)ᵀ(AB) = I Where I is the identity matrix

Correct question : If A & B are orthogonal , then (AB)ᵀ(AB) is equal to

Given :

A & B are orthogonal

To find :

The value of (AB)ᵀ(AB)

Concept :

A Matrix M is said to be orthogonal iff MᵀM = MMᵀ = I

Where Mᵀ is transpose of M

Solution :

Step 1 of 2 :

Write down the given data

Here it is given that A & B are orthogonal

∴ AAᵀ = AᵀA = I & BBᵀ = BᵀB = I

Where Aᵀ and Bᵀ are transpose of A & B respectively

Step 2 of 2 :

Find the value of (AB)ᵀ(AB)

(AB)ᵀ(AB)

= BᵀAᵀAB [ ∵ (AB)ᵀ = BᵀAᵀ ]

= Bᵀ(AᵀA)B

= Bᵀ(I)B [ ∵ AᵀA = I ]

= Bᵀ(IB)

= BᵀB [ ∵ IB = B ]

= I [ ∵ AᵀA = I ]

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