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 the problem, we need to find the value of \( \lambda \) in the matrix \( A \) given that it is the adjoint of matrix \( B \) and the determinant of \( B \) is \( |B| = 9 \).

### Step 1: Understand the relationship between the adjoint and the determinant
The relationship between the determinant of a matrix \( A \) and its adjoint \( B \) is given by the formula:
\[ |A| = |B|^{n-1} \]
where \( n \) is the order of the matrix. Here, since \( A \) is a \( 3 \times 3 \) matrix, \( n = 3 \).

### Step 2: Substitute the values into the formula
Given \( |B| = 9 \), we can substitute this into the formula:
\[ |A| = |B|^{3-1} = |B|^2 = 9^2 = 81 \]

### Step 3: Calculate the determinant of matrix \( A \)
Now we need to calculate 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 \( 3 \times 3 \) matrix:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is represented as:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix \( A \):
- \( a = 1, b = -2, c = 1 \)
- \( d = 2, e = \lambda, f = -2 \)
- \( g = 1, h = 3, i = -3 \)

### Step 4: Calculate each part of the determinant
1. Calculate \( ei - fh \):
\[ ei - fh = \lambda \cdot (-3) - (-2) \cdot 3 = -3\lambda + 6 \]

2. Calculate \( di - fg \):
\[ di - fg = 2 \cdot (-3) - (-2) \cdot 1 = -6 + 2 = -4 \]

3. Calculate \( dh - eg \):
\[ dh - eg = 2 \cdot 3 - \lambda \cdot 1 = 6 - \lambda \]

### Step 5: Substitute into the determinant formula
Now substituting these values into the determinant formula:
\[ |A| = 1(-3\lambda + 6) - (-2)(-4) + 1(6 - \lambda) \]
\[ |A| = -3\lambda + 6 - 8 + 6 - \lambda \]
\[ |A| = -4\lambda + 4 \]

### Step 6: Set the determinant equal to 81
Now we set the determinant equal to 81:
\[ -4\lambda + 4 = 81 \]

### Step 7: Solve for \( \lambda \)
Rearranging gives:
\[ -4\lambda = 81 - 4 \]
\[ -4\lambda = 77 \]
\[ \lambda = -\frac{77}{4} \]

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

### Final Answer
The value of \( \lambda \) is \( -\frac{77}{4} \).

---