https://file.aiquickdraw.com/mathbot/file/1726862428488q2sx0ahk.jpg

To determine if the matrix represented in the image is unitary, we need to recall that a matrix \( U \) is unitary if it satisfies the condition:

\[
U^\dagger U = U U^\dagger = I
\]

where \( U^\dagger \) is the conjugate transpose of \( U \), and \( I \) is the identity matrix.

Let's start by analyzing the matrix given in the image. The expression in the image represents a 2x2 matrix:

\[
\frac{1}{2} \begin{bmatrix} 1+i & -1+i \\ 1+i & 1-i \end{bmatrix}
\]

This is the matrix \( U \).

### Step 1: Find \( U^\dagger \) (the conjugate transpose of \( U \))

To find \( U^\dagger \), we take the transpose of \( U \) and then take the complex conjugate of each element.

1. **Transpose** the matrix \( U \):

\[
U^T = \begin{bmatrix} 1+i & 1+i \\ -1+i & 1-i \end{bmatrix}
\]

2. **Complex conjugate** of each element:

\[
U^\dagger = \begin{bmatrix} 1-i & 1-i \\ -1-i & 1+i \end{bmatrix}
\]

### Step 2: Multiply \( U^\dagger U \)

Now, we need to multiply \( U^\dagger \) by \( U \):

\[
U^\dagger U = \frac{1}{2} \begin{bmatrix} 1-i & 1-i \\ -1-i & 1+i \end{bmatrix} \cdot \frac{1}{2} \begin{bmatrix} 1+i & -1+i \\ 1+i & 1-i \end{bmatrix}
\]

We will perform the matrix multiplication step by step.

#### Top-left element:
\[
(1-i)(1+i) + (1-i)(1+i) = 2 + 2 = 4
\]

#### Top-right element:
\[
(1-i)(-1+i) + (1-i)(1-i) = 0 + 0 = 0
\]

#### Bottom-left element:
\[
(-1-i)(1+i) + (1+i)(1+i) = 0 + 0 = 0
\]

#### Bottom-right element:
\[
(-1-i)(-1+i) + (1+i)(1-i) = 2 + 2 = 4
\]

Thus,

\[
U^\dagger U = \frac{1}{4} \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

This is the identity matrix \( I \).

### Conclusion:
Since \( U^\dagger U = I \), the matrix \( U \) is unitary.

Let me know if you want further details or have any questions!

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Here are some related questions you might want to explore:

1. What are the properties of a unitary matrix?
2. How does the determinant of a unitary matrix relate to its eigenvalues?
3. What is the physical significance of unitary matrices in quantum mechanics?
4. How do we verify if a matrix is Hermitian?
5. Can a matrix be both unitary and orthogonal?

**Tip:** The product of two unitary matrices is also unitary, making them very useful in constructing quantum gates and transformations.

Explore how to prove a matrix is unitary by applying the formula U^\dagger U = I. This solution involves matrix multiplication, complex conjugates, and the unitary matrix theorem, suitable for advanced high school or college-level students.

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