Let `|(a,l,m),(l,b,n),(m,n,c)||(bc-n^2,mn-lc,ln-bm),(mn-lc,ac-m^2,ml-an),(ln-bm,lm-an,ab-l^2)|=64.` If the value of `|(2a+3l,3l+5m,5m+4a),(2l+3b,3b+5n,5n+4l),(2m+3n,3n+5c,5c+4m)|=lambda` then `[lambda/2]` equals

Options

A. `180`
B. `240`
C. `360`
D. `480`

Explanation

To solve the given problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Define the Determinant
Let
\[ D = \begin{vmatrix} a & l & m \\ l & b & n \\ m & n & c \end{vmatrix} \]
We are given that \( |D| = 64 \).

### Step 2: Find the Value of the Determinant's Adjoint
The adjoint of a matrix \( D \) of order \( n \) has a determinant given by:
\[ |adj(D)| = |D|^{n-1} \]
Here, \( n = 3 \), so:
\[ |adj(D)| = |D|^{3-1} = |D|^2 \]
Since \( |D| = 64 \), we have:
\[ |adj(D)| = 64^2 = 4096 \]

### Step 3: Relate the Determinants
The determinant of the second matrix given in the problem is:
\[ \lambda = \begin{vmatrix} 2a + 3l & 3l + 5m & 5m + 4a \\ 2l + 3b & 3b + 5n & 5n + 4l \\ 2m + 3n & 3n + 5c & 5c + 4m \end{vmatrix} \]
We can express this determinant in terms of \( D \).

### Step 4: Factor Out Common Terms
We can factor out constants from each column:
- From the first column, factor out \( 2 \).
- From the second column, factor out \( 3 \).
- From the third column, factor out \( 5 \).

Thus, we can write:
\[ \lambda = 2 \cdot 3 \cdot 5 \cdot \begin{vmatrix} a & l & m \\ l & b & n \\ m & n & c \end{vmatrix} + \text{other terms} \]
The "other terms" will also involve the determinant \( D \) in a similar manner.

### Step 5: Calculate the Value of Lambda
From the structure of the determinant, we can conclude that:
\[ \lambda = (2 \cdot 3 \cdot 5 + 3 \cdot 5 \cdot 4) \cdot |D| \]
Calculating the coefficients:
\[ 2 \cdot 3 \cdot 5 = 30 \]
\[ 3 \cdot 5 \cdot 4 = 60 \]
Thus:
\[ \lambda = (30 + 60) \cdot |D| = 90 \cdot 64 = 5760 \]

### Step 6: Find the Greatest Integer of Lambda/2
Now we need to find \( \left\lfloor \frac{\lambda}{2} \right\rfloor \):
\[ \frac{\lambda}{2} = \frac{5760}{2} = 2880 \]
Thus, the greatest integer of \( \frac{\lambda}{2} \) is:
\[ \left\lfloor 2880 \right\rfloor = 2880 \]

### Final Answer
The value of \( \left\lfloor \frac{\lambda}{2} \right\rfloor \) is \( 2880 \).