The element in the `i^(th)` row and the `j^(th)` column of a determinant of third order is equal to `2(i+j)`. What is the value of the determinant?

Options

A. 0
B. 2
C. 4
D. 6

Answer

To find the value of the determinant of the given matrix where the element in the \(i^{th}\) row and \(j^{th}\) column is defined as \(2(i+j)\), we will follow these steps:

### Step 1: Construct the Matrix
The matrix \(A\) of order 3 can be constructed as follows:

\[ A = \begin{bmatrix} 2(1+1) & 2(1+2) & 2(1+3) \\ 2(2+1) & 2(2+2) & 2(2+3) \\ 2(3+1) & 2(3+2) & 2(3+3) \end{bmatrix} \]

Calculating the elements:
- For \(i=1\):
- \(A_{11} = 2(1+1) = 4\)
- \(A_{12} = 2(1+2) = 6\)
- \(A_{13} = 2(1+3) = 8\)

- For \(i=2\):
- \(A_{21} = 2(2+1) = 6\)
- \(A_{22} = 2(2+2) = 8\)
- \(A_{23} = 2(2+3) = 10\)

- For \(i=3\):
- \(A_{31} = 2(3+1) = 8\)
- \(A_{32} = 2(3+2) = 10\)
- \(A_{33} = 2(3+3) = 12\)

Thus, the matrix \(A\) is:

\[ A = \begin{bmatrix} 4 & 6 & 8 \\ 6 & 8 & 10 \\ 8 & 10 & 12 \end{bmatrix} \]

### Step 2: Perform Column Operations
Next, we will perform column operations to simplify the determinant. We can subtract the second column from the first and the third column:

\[ C_1 \rightarrow C_1 - C_2 \quad \text{and} \quad C_3 \rightarrow C_3 - C_2 \]

This gives us the new matrix:

\[ A = \begin{bmatrix} 4 - 6 & 6 & 8 - 6 \\ 6 - 8 & 8 & 10 - 8 \\ 8 - 10 & 10 & 12 - 10 \end{bmatrix} \]

Calculating the new elements:

\[ A = \begin{bmatrix} -2 & 6 & 2 \\ -2 & 8 & 2 \\ -2 & 10 & 2 \end{bmatrix} \]

### Step 3: Factor Out Common Elements
Now we can factor out \(-2\) from the first column:

\[ A = -2 \begin{bmatrix} 1 & 6 & 2 \\ 1 & 8 & 2 \\ 1 & 10 & 2 \end{bmatrix} \]

### Step 4: Check for Row Similarity
Notice that the first column now has identical elements (1, 1, 1). Since two rows of the matrix are identical, the determinant of this matrix is 0.

### Step 5: Calculate the Determinant
Thus, the determinant of the original matrix is:

\[ \text{det}(A) = -2 \cdot 0 = 0 \]

### Final Answer
The value of the determinant is \( \boxed{0} \).

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