If `A=[(1,-2,1),(2,lambda,-2),(1,3,-3)]` be the adjoint matrix of matrix B such that `|B|=9`, then the value of `lambda` is equal to

Options

A. 1
B. `(-77)/(4)`
C. `(23)/(2)`
D. `(-39)/(2)`

Answer

To solve for the value of \( \lambda \) in the matrix \( A = \begin{pmatrix} 1 & -2 & 1 \\ 2 & \lambda & -2 \\ 1 & 3 & -3 \end{pmatrix} \), given that \( A \) is the adjoint of matrix \( B \) and \( |B| = 9 \), we can follow these steps:

### Step 1: Use the property of determinants
The property of determinants states that if \( A \) is the adjoint of \( B \), then:
\[ |A| = |B|^2 \]
Given \( |B| = 9 \), we have:
\[ |A| = 9^2 = 81 \]

### Step 2: Calculate the determinant of matrix \( A \)
We need to compute the determinant of matrix \( A \):
\[ A = \begin{pmatrix} 1 & -2 & 1 \\ 2 & \lambda & -2 \\ 1 & 3 & -3 \end{pmatrix} \]
Using the determinant formula for a 3x3 matrix:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( a, b, c \) are the elements of the first row, and \( d, e, f, g, h, i \) are the elements of the remaining rows.

Substituting the values:
\[ |A| = 1 \cdot (\lambda \cdot (-3) - (-2) \cdot 3) - (-2) \cdot (2 \cdot (-3) - (-2) \cdot 1) + 1 \cdot (2 \cdot 3 - \lambda \cdot 1) \]

### Step 3: Simplify the determinant expression
Calculating each term:
1. First term:
\[ 1 \cdot (-3\lambda + 6) = -3\lambda + 6 \]
2. Second term:
\[ -(-2) \cdot (-6 + 2) = 2 \cdot (-4) = -8 \]
3. Third term:
\[ 1 \cdot (6 - \lambda) = 6 - \lambda \]

Combining these:
\[ |A| = -3\lambda + 6 - 8 + 6 - \lambda \]
\[ |A| = -4\lambda + 4 \]

### Step 4: Set the determinant equal to 81
Now we set the determinant equal to 81:
\[ -4\lambda + 4 = 81 \]

### Step 5: Solve for \( \lambda \)
Rearranging gives:
\[ -4\lambda = 81 - 4 \]
\[ -4\lambda = 77 \]
\[ \lambda = -\frac{77}{4} \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \lambda = -\frac{77}{4} \]