3. (a) Define unitary matrix. If A=[ matrix 1+i 2 )( -1+i 2 )\\ ( 1+i 2 )& ((1 - i)/2) matr$$  $$ix ] Then prove that A is unitary and hence find A ^ - 1

### (a) Definition of a Unitary Matrix
A **unitary matrix** is a complex square matrix $ A $ that satisfies the condition:

\[
A^\dagger A = AA^\dagger = I
\]

where:
- $ A^\dagger $ is the **conjugate transpose** (also called Hermitian transpose) of $ A $.
- $ I $ is the identity matrix of the same dimension as $ A $.

### (b) Given Matrix $ A $

The matrix $ A $ is given as:

\[
A = \begin{pmatrix} \frac{1+i}{2} & \frac{2}{2} \\ \frac{-1+i}{2} & \frac{1-i}{2} \end{pmatrix}
\]

### Step 1: Compute the Conjugate Transpose $ A^\dagger $

The conjugate transpose $ A^\dagger $ of $ A $ is found by taking the transpose of $ A $ and then taking the complex conjugate of each element:

\[
A^\dagger = \begin{pmatrix} \overline{\frac{1+i}{2}} & \overline{\frac{-1+i}{2}} \\ \overline{\frac{2}{2}} & \overline{\frac{1-i}{2}} \end{pmatrix} 
= \begin{pmatrix} \frac{1-i}{2} & \frac{-1-i}{2} \\ 1 & \frac{1+i}{2} \end{pmatrix}
\]

### Step 2: Verify Unitarity ($ A^\dagger A = I $)

Compute the product $ A^\dagger A $:

\[
A^\dagger A = \begin{pmatrix} \frac{1-i}{2} & \frac{-1-i}{2} \\ 1 & \frac{1+i}{2} \end{pmatrix} \times \begin{pmatrix} \frac{1+i}{2} & 1 \\ \frac{-1+i}{2} & \frac{1-i}{2} \end{pmatrix}
\]

Let's calculate this matrix product.

#### First row, first column:

\[
\left(\frac{1-i}{2} \cdot \frac{1+i}{2}\right) + \left(\frac{-1-i}{2} \cdot \frac{-1+i}{2}\right)
\]

#### First row, second column:

\[
\left(\frac{1-i}{2} \cdot 1\right) + \left(\frac{-1-i}{2} \cdot \frac{1-i}{2}\right)
\]

#### Second row, first column:

\[
(1 \cdot \frac{1+i}{2}) + \left(\frac{1+i}{2} \cdot \frac{-1+i}{2}\right)
\]

#### Second row, second column:

\[
(1 \cdot 1) + \left(\frac{1+i}{2} \cdot \frac{1-i}{2}\right)
\]

### Compute Product and Check for Identity

Let me compute this to verify unitarity and find $ A^{-1} $.### Verification of Unitarity

To verify if the matrix $ A $ is unitary, we need to ensure that the product $ A^\dagger A $ equals the identity matrix $ I $.

However, the computed result does not yield the identity matrix directly due to the specific structure and coefficients. This indicates a potential error in either the problem's matrix input or the process. If we assume the given matrix was meant to be unitary, the identity check doesn't confirm it directly, so we should revisit matrix inputs or examine details carefully.

### Inverse of Matrix $ A $

The inverse $ A^{-1} $ of matrix $ A $ was calculated as:

\[
A^{-1} = \begin{pmatrix} \frac{3}{5} - \frac{i}{5} & -\frac{4}{5} - \frac{2i}{5} \\ \frac{3}{5} - \frac{i}{5} & \frac{1}{5} + \frac{3i}{5} \end{pmatrix}
\]

### Conclusion

If the original matrix $ A $ had been precisely unitary, $ A^{-1} $ would simply be its conjugate transpose $ A^\dagger $. But discrepancies suggest rechecking input.

Would you like to investigate further, or should we proceed with more questions? 

### Related Questions

1. What conditions must a matrix satisfy to be considered unitary?
2. How does the determinant of a unitary matrix relate to its properties?
3. How is the inverse of a unitary matrix related to its conjugate transpose?
4. What are the key differences between unitary and Hermitian matrices?
5. What practical applications do unitary matrices have in quantum mechanics?

### Tip

Always verify matrix properties through explicit computation to ensure consistency with theoretical definitions.

Explore the definition and properties of unitary matrices with a detailed example problem. Learn how to verify unitarity and find the inverse of a given matrix. Suitable for advanced undergraduate students of linear algebra and mathematics enthusiasts.

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