If any two rows (or columns) of a determinant are identical, then the value of the determinant is zero.

Answer

To prove that if any two rows (or columns) of a determinant are identical, then the value of the determinant is zero, we can follow these steps:

### Step 1: Define the Determinant
Let \( A \) be a determinant defined as follows:

\[ A = \begin{vmatrix} 2 & 4 & 6 \\ 2 & 4 & 6 \\ 1 & 3 & 2 \end{vmatrix} \]

### Step 2: Modify One of the Rows
We can perform a row operation on the determinant. Specifically, we can subtract the first row from the second row:

\[ R_2 \rightarrow R_2 - R_1 \]

This gives us the new determinant:

\[ A = \begin{vmatrix} 2 & 4 & 6 \\ 0 & 0 & 0 \\ 1 & 3 & 2 \end{vmatrix} \]

### Step 3: Expand the Determinant
Now, we can expand this determinant along the second row (which is now all zeros):

\[ A = 2 \cdot \begin{vmatrix} 0 & 0 \\ 3 & 2 \end{vmatrix} - 0 + 0 \]

### Step 4: Calculate the Value
Since the second row is all zeros, the determinant evaluates to zero:

\[ A = 2 \cdot 0 - 0 + 0 = 0 \]

### Step 5: Conclusion
Thus, we have shown that if two rows of a determinant are identical, the value of the determinant is zero.