Write the value of the determinant `|[2, 3, 4],[ 5, 6, 8],[ 6x,9x, 12 x]|`

Options

A. 0
B. 10
C. 10x
D. 100x

Explanation

To find the value of the determinant of the matrix

\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{bmatrix} \]

we will follow these steps:

### Step 1: Factor out the common term from the third row.
The third row of the matrix contains a common factor of \(3x\). We can factor this out:

\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 3x(2) & 3x(3) & 3x(4) \end{bmatrix} \]

This simplifies to:

\[ A = 3x \begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 2 & 3 & 4 \end{bmatrix} \]

### Step 2: Observe the rows of the matrix.
Now, we can see that the first row and the third row are identical:

\[ \begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 2 & 3 & 4 \end{bmatrix} \]

### Step 3: Use the property of determinants.
A property of determinants states that if two rows (or columns) of a matrix are identical, then the determinant of that matrix is zero.

### Step 4: Conclude the value of the determinant.
Since we have two identical rows in the matrix, the determinant of the matrix is:

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

### Final Answer:
Thus, the value of the determinant is:

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

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