If `|(-12,0,lamda),(0,2,-1),(2,1,15)| = -360`, then the value of `lamda` is

Options

A. `-1`
B. `-2`
C. `-3`
D. 4

Answer

To solve the determinant equation \(|(-12, 0, \lambda), (0, 2, -1), (2, 1, 15)| = -360\), we will follow these steps:

### Step 1: Write the determinant
We have the determinant:
\[ D = \begin{vmatrix} -12 & 0 & \lambda \\ 0 & 2 & -1 \\ 2 & 1 & 15 \end{vmatrix} \]

### Step 2: Expand the determinant along the first row
Using the formula for the determinant of a 3x3 matrix, we expand along the first row:
\[ D = -12 \begin{vmatrix} 2 & -1 \\ 1 & 15 \end{vmatrix} - 0 + \lambda \begin{vmatrix} 0 & 2 \\ 2 & 1 \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants
Now we calculate the two 2x2 determinants:
1. For \(\begin{vmatrix} 2 & -1 \\ 1 & 15 \end{vmatrix}\):
\[ = (2 \cdot 15) - (-1 \cdot 1) = 30 + 1 = 31 \]

2. For \(\begin{vmatrix} 0 & 2 \\ 2 & 1 \end{vmatrix}\):
\[ = (0 \cdot 1) - (2 \cdot 2) = 0 - 4 = -4 \]

### Step 4: Substitute back into the determinant equation
Substituting these values back into the determinant equation:
\[ D = -12 \cdot 31 + \lambda \cdot (-4) \]
\[ D = -372 - 4\lambda \]

### Step 5: Set the determinant equal to -360
Now we set the determinant equal to -360:
\[ -372 - 4\lambda = -360 \]

### Step 6: Solve for \(\lambda\)
Rearranging the equation gives:
\[ -4\lambda = -360 + 372 \]
\[ -4\lambda = 12 \]
\[ \lambda = \frac{12}{-4} = -3 \]

### Conclusion
Thus, the value of \(\lambda\) is \(-3\).