Mathematics Exam > Mathematics Questions > Which of the following(s) is/are correct?a)If...
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Which of the following(s) is/are correct?
- a)If A is unitary matrix, then A'is also unitary matrix.
- b)Inverse of unitary matrix is unitary matrix.
- c)Inverse of unitary matrix is diagonal matrix.
- d)None of these
Correct answer is option 'A,B'. Can you explain this answer?
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Which of the following(s) is/are correct?a)If A is unitary matrix, the...
Answer:
Statement a: If A is a unitary matrix, then A is also a unitary matrix.
Statement b: The inverse of a unitary matrix is a unitary matrix.
Explanation:
A matrix is said to be unitary if its conjugate transpose is equal to its inverse. In other words, a matrix A is unitary if A*A^H = I, where A^H is the conjugate transpose of A and I is the identity matrix.
Statement a: If A is a unitary matrix, then A is also a unitary matrix.
This statement is trivially true. If A is a unitary matrix, then by definition, A*A^H = I. But A^H*A = (A*A^H)^H = I^H = I. Therefore, A is also a unitary matrix.
Statement b: The inverse of a unitary matrix is a unitary matrix.
To prove this statement, let's consider a unitary matrix A. We need to show that its inverse, denoted by A^(-1), is also a unitary matrix.
Since A is unitary, we have A*A^H = I. Now let's consider the product of A^H*A^(-1):
(A^H)*(A^(-1)) = (A^(-1))^H*(A^H)^H = (A^(-1))^H*A.
By substituting the value of A*A^H from the given condition, we have:
(A^(-1))^H*A = (A^(-1))^H*(A^H)*(A*A^H) = (A^(-1))^H*(A^H)*I = (A^(-1))^H*I = (A^(-1))^H.
Therefore, (A^H)*(A^(-1)) = (A^(-1))^H, which implies that A^(-1) is the inverse of A.
Now, let's verify if A^(-1) is a unitary matrix:
(A^(-1))*(A^(-1))^H = A^(-1)*(A^H)^(-1) = (A*A^H)^(-1) = I^(-1) = I.
Hence, A^(-1) is a unitary matrix.
Conclusion:
Based on the above explanations, both statement a and statement b are correct.
Statement a: If A is a unitary matrix, then A is also a unitary matrix.
Statement b: The inverse of a unitary matrix is a unitary matrix.
Explanation:
A matrix is said to be unitary if its conjugate transpose is equal to its inverse. In other words, a matrix A is unitary if A*A^H = I, where A^H is the conjugate transpose of A and I is the identity matrix.
Statement a: If A is a unitary matrix, then A is also a unitary matrix.
This statement is trivially true. If A is a unitary matrix, then by definition, A*A^H = I. But A^H*A = (A*A^H)^H = I^H = I. Therefore, A is also a unitary matrix.
Statement b: The inverse of a unitary matrix is a unitary matrix.
To prove this statement, let's consider a unitary matrix A. We need to show that its inverse, denoted by A^(-1), is also a unitary matrix.
Since A is unitary, we have A*A^H = I. Now let's consider the product of A^H*A^(-1):
(A^H)*(A^(-1)) = (A^(-1))^H*(A^H)^H = (A^(-1))^H*A.
By substituting the value of A*A^H from the given condition, we have:
(A^(-1))^H*A = (A^(-1))^H*(A^H)*(A*A^H) = (A^(-1))^H*(A^H)*I = (A^(-1))^H*I = (A^(-1))^H.
Therefore, (A^H)*(A^(-1)) = (A^(-1))^H, which implies that A^(-1) is the inverse of A.
Now, let's verify if A^(-1) is a unitary matrix:
(A^(-1))*(A^(-1))^H = A^(-1)*(A^H)^(-1) = (A*A^H)^(-1) = I^(-1) = I.
Hence, A^(-1) is a unitary matrix.
Conclusion:
Based on the above explanations, both statement a and statement b are correct.
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