A square matrix A is called orthogonal if
Where A' is the transpose of A.

Options

A. `A=A^(2)`
B. `A'=A^(-1)`
C. `A=A^(-1)`
D. `A= A'`

Explanation

To determine whether a square matrix \( A \) is orthogonal, we need to verify the condition that \( A^T A = I \), where \( A^T \) is the transpose of \( A \) and \( I \) is the identity matrix. Here's a step-by-step solution:

### Step 1: Understanding the Definition
An orthogonal matrix is defined such that the product of the matrix and its transpose equals the identity matrix. Mathematically, this is expressed as:
\[ A^T A = I \]

### Step 2: Transpose of the Matrix
The transpose of a matrix \( A \) is obtained by flipping the matrix over its diagonal, which means the row and column indices are switched. For example, if:
\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
then the transpose \( A^T \) is:
\[ A^T = \begin{pmatrix} a & c \\ b & d \end{pmatrix} \]

### Step 3: Multiplying the Matrix by its Transpose
Next, we compute the product \( A^T A \). Using our example:
\[ A^T A = \begin{pmatrix} a & c \\ b & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
This results in:
\[ A^T A = \begin{pmatrix} a^2 + c^2 & ab + cd \\ ab + cd & b^2 + d^2 \end{pmatrix} \]

### Step 4: Setting the Product Equal to the Identity Matrix
For \( A \) to be orthogonal, we need:
\[ A^T A = I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
This gives us the equations:
1. \( a^2 + c^2 = 1 \)
2. \( b^2 + d^2 = 1 \)
3. \( ab + cd = 0 \)

### Step 5: Verifying the Conditions
To confirm that \( A \) is orthogonal, we need to check if the above conditions hold true for the elements of \( A \). If they do, then \( A \) is indeed an orthogonal matrix.

### Conclusion
Thus, we conclude that a square matrix \( A \) is orthogonal if:
\[ A^T A = I \quad \text{or equivalently} \quad A^T = A^{-1} \]