The product of two Unitary matrices is a ________ matrixa)Unitb)Orthogonalc)Unitaryd)None of the aboveCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Unitary Matrices Product

Unitary matrices are matrices that satisfy the property: U*U^H = I, where U^H is the conjugate transpose of U and I is the identity matrix.

Product of Two Unitary Matrices

When you multiply two unitary matrices together, the result is also a unitary matrix. This can be proven as follows:

Let A and B be two unitary matrices. The product of A and B is AB.
Taking the conjugate transpose of AB, we get (AB)^H = B^H * A^H.

Now, to show that AB is unitary, we need to check if (AB)(AB)^H = I.
(AB)(AB)^H = AB * (B^H * A^H) = A(BB^H)A^H.

Since A and B are unitary matrices, A*A^H = I and B*B^H = I.
Therefore, A(BB^H)A^H = AIA^H = AA^H = I.

Hence, the product of two unitary matrices is also a unitary matrix.

Answer

the product of two unitary matrices is always unitary.

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