If A' is the transpose of a square matrix A, then:

Options

A. `|A| ne |A'|`
B. |A| = |A'|
C. |A|+ |A'|=0
D. |A| = |A'| only, when A is symmetric

Answer

To solve the problem regarding the relationship between a square matrix \( A \) and its transpose \( A' \) (or \( A^T \)), we need to analyze the determinants of both matrices. Here’s a step-by-step solution:

### Step 1: Define the square matrix \( A \)
Let’s consider a square matrix \( A \):
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]

### Step 2: Calculate the determinant of \( A \)
The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated as:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
\[ \text{det}(A) = (1)(4) - (2)(3) = 4 - 6 = -2 \]

### Step 3: Find the transpose of matrix \( A \)
The transpose of matrix \( A \), denoted as \( A' \) or \( A^T \), is obtained by swapping the rows and columns:
\[ A' = A^T = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \]

### Step 4: Calculate the determinant of \( A' \)
Now, we calculate the determinant of the transpose \( A' \):
\[ \text{det}(A') = (1)(4) - (3)(2) = 4 - 6 = -2 \]

### Step 5: Compare the determinants
From our calculations, we have:
\[ \text{det}(A) = -2 \quad \text{and} \quad \text{det}(A') = -2 \]
This shows that:
\[ \text{det}(A) = \text{det}(A') \]

### Conclusion
The determinants of a square matrix and its transpose are equal:
\[ \text{det}(A) = \text{det}(A^T) \]
Thus, the correct option is that the determinants of \( A \) and \( A' \) are equal.