Let `A=[a_(ij)]_(3xx3)` be a square matrix such that `A A^(T)=4I, |A| lt 0`. If `|(a_(11)+4,a_(12),a_(13)),(a_(21),a_(22)+4,a_(23)),(a_(31),a_(32),a_(33)+4)|=5lambda|A+I|.` Then `lambda` is equal to

Options

A. `(4)/(5)`
B. `-(4)/(5)`
C. `(8)/(5)`
D. `-(8)/(5)`

Answer

To solve the problem step by step, we start with the given conditions about the matrix \( A \).

### Step 1: Analyze the given conditions
We know that:
1. \( A A^T = 4I \)
2. \( |A| < 0 \)

### Step 2: Take the determinant of both sides of the first condition
Taking the determinant of both sides of the equation \( A A^T = 4I \):
\[ |A A^T| = |4I| \]
Using the property of determinants, we have:
\[ |A| \cdot |A^T| = |A|^2 = 4^3 = 64 \]
Thus, we can write:
\[ |A|^2 = 64 \]

### Step 3: Solve for the determinant of \( A \)
Taking the square root of both sides gives:
\[ |A| = \pm 8 \]
Since we are given that \( |A| < 0 \), we conclude:
\[ |A| = -8 \]

### Step 4: Use the second condition
We need to evaluate the determinant of the matrix:
\[ \begin{vmatrix} a_{11} + 4 & a_{12} & a_{13} \\ a_{21} & a_{22} + 4 & a_{23} \\ a_{31} & a_{32} & a_{33} + 4 \end{vmatrix} \]
This can be rewritten as:
\[ \begin{vmatrix} A + 4I \end{vmatrix} \]
where \( 4I \) is a diagonal matrix with 4 on the diagonal.

### Step 5: Apply the determinant property
Using the property of determinants, we have:
\[ |A + 4I| = |A| + 4 \cdot \text{trace}(A) + 4^2 \cdot \text{other terms} \]
However, we can use the fact that:
\[ |A + 4I| = |A| + 4 \cdot |A| + 16 \]
This gives us:
\[ |A + 4I| = |A| + 4 \cdot |A| + 16 \]

### Step 6: Set up the equation
From the problem statement, we know:
\[ |(A + 4I)| = 5\lambda |A + I| \]

### Step 7: Substitute the known values
Substituting \( |A| = -8 \):
\[ |-8 + 4(-8) + 16| = 5\lambda |-8 + 1| \]
This simplifies to:
\[ |-8 - 32 + 16| = 5\lambda |-7| \]
Thus:
\[ |-24| = 5\lambda \cdot 7 \]
This gives:
\[ 24 = 35\lambda \]

### Step 8: Solve for \( \lambda \)
Now, we can solve for \( \lambda \):
\[ \lambda = \frac{24}{35} \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \lambda = \frac{24}{35} \]