Matrix `A_(lamda)=[(lamda, lamda-1),(lamda-1,lamda)], lamda in N`
The value of `|A_(1)|+|A_(2)|+…….+|A_(300)|` is

Options

A. `(299)^(2)`
B. `(300)^(2)`
C. `(301)^(2)`
D. None of these

Answer

To solve the problem, we need to calculate the sum of the determinants of the matrix \( A_{\lambda} \) for \( \lambda \) ranging from 1 to 300. The matrix is given as:

\[ A_{\lambda} = \begin{pmatrix} \lambda & \lambda - 1 \\ \lambda - 1 & \lambda \end{pmatrix} \]

### Step 1: Calculate the Determinant of \( A_{\lambda} \)

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula:

\[ \text{det}(A) = ad - bc \]

For our matrix \( A_{\lambda} \):

- \( a = \lambda \)
- \( b = \lambda - 1 \)
- \( c = \lambda - 1 \)
- \( d = \lambda \)

Thus, the determinant is:

\[ |A_{\lambda}| = \lambda \cdot \lambda - (\lambda - 1)(\lambda - 1) \]

Expanding this gives:

\[ |A_{\lambda}| = \lambda^2 - (\lambda^2 - 2\lambda + 1) = \lambda^2 - \lambda^2 + 2\lambda - 1 = 2\lambda - 1 \]

### Step 2: Calculate the Sum of Determinants from 1 to 300

Now we need to calculate the sum:

\[ |A_1| + |A_2| + |A_3| + \ldots + |A_{300}| \]

Substituting the determinant we found:

\[ |A_1| + |A_2| + \ldots + |A_{300}| = (2 \cdot 1 - 1) + (2 \cdot 2 - 1) + (2 \cdot 3 - 1) + \ldots + (2 \cdot 300 - 1) \]

This simplifies to:

\[ = (2 - 1) + (4 - 1) + (6 - 1) + \ldots + (600 - 1) \]

### Step 3: Simplify the Expression

The sum can be separated into two parts:

\[ = (2 + 4 + 6 + \ldots + 600) - (1 + 1 + \ldots + 1) \]

The first part is the sum of the first 300 even numbers, which can be calculated using the formula for the sum of the first \( n \) even numbers:

\[ \text{Sum of first } n \text{ even numbers} = n(n + 1) \]

For \( n = 300 \):

\[ \text{Sum} = 300 \cdot (300 + 1) = 300 \cdot 301 = 90300 \]

The second part is simply \( 300 \) (since we are subtracting 1 for each of the 300 terms).

### Step 4: Final Calculation

Now we can calculate the total:

\[ = 90300 - 300 = 90000 \]

### Conclusion

Thus, the value of \( |A_1| + |A_2| + \ldots + |A_{300}| \) is:

\[ \boxed{90000} \]