The set of natural numbers is divided into array of rows and columns in the form of matrices as `A_(1)=[1], A_(2)=[(2,3),(4,5)], A_(3)=[(6,7,8),(9,10,11),(12,13,14)]` and so on. Let the trace of `A_(10)` be `lambda` . Find unit digit of `lambda` ?

Answer

To find the unit digit of the trace of the matrix \( A_{10} \), we will follow these steps systematically:

### Step 1: Understand the structure of matrices
The matrices are defined as follows:
- \( A_1 = [1] \)
- \( A_2 = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \)
- \( A_3 = \begin{bmatrix} 6 & 7 & 8 \\ 9 & 10 & 11 \\ 12 & 13 & 14 \end{bmatrix} \)

From this, we can see that \( A_n \) is an \( n \times n \) matrix where the first element of \( A_n \) is the sum of the first \( n-1 \) natural numbers plus 1.

### Step 2: Find the first element of \( A_{10} \)
The first element \( a_{11} \) of \( A_n \) can be expressed as:
\[ a_{11} = 1 + 1^2 + 2^2 + 3^2 + \ldots + (n-1)^2 \]
For \( n = 10 \):
\[ a_{11} = 1 + 1^2 + 2^2 + 3^2 + \ldots + 9^2 \]
Using the formula for the sum of squares of the first \( n \) natural numbers:
\[ \text{Sum of squares} = \frac{n(n+1)(2n+1)}{6} \]
Substituting \( n = 9 \):
\[ \text{Sum of squares} = \frac{9(9+1)(2 \cdot 9 + 1)}{6} = \frac{9 \cdot 10 \cdot 19}{6} \]

### Step 3: Calculate \( a_{11} \)
Calculating the above expression:
\[ = \frac{1710}{6} = 285 \]
Thus,
\[ a_{11} = 1 + 285 = 286 \]

### Step 4: Find the diagonal elements of \( A_{10} \)
The diagonal elements \( a_{22}, a_{33}, \ldots, a_{10,10} \) can be found using the common difference in the arithmetic progression formed by the elements of the matrix. The common difference \( d \) for \( A_n \) is \( n + 1 \).

For \( A_{10} \):
- First element \( a_{11} = 286 \)
- Common difference \( d = 11 \)

The diagonal elements can be expressed as:
- \( a_{22} = a_{11} + 11 = 286 + 11 = 297 \)
- \( a_{33} = a_{11} + 2 \cdot 11 = 286 + 22 = 308 \)
- Continuing this, we find:
\[ a_{kk} = 286 + (k-1) \cdot 11 \quad \text{for } k = 1, 2, \ldots, 10 \]

### Step 5: Calculate the trace \( \lambda \)
The trace \( \lambda \) is the sum of the diagonal elements:
\[ \lambda = a_{11} + a_{22} + a_{33} + \ldots + a_{10,10} \]
This can be expressed as:
\[ \lambda = \sum_{k=1}^{10} (286 + (k-1) \cdot 11) \]
This simplifies to:
\[ \lambda = 10 \cdot 286 + 11 \cdot \sum_{k=0}^{9} k = 10 \cdot 286 + 11 \cdot \frac{9 \cdot 10}{2} \]
Calculating:
\[ = 2860 + 11 \cdot 45 = 2860 + 495 = 3355 \]

### Step 6: Find the unit digit of \( \lambda \)
The unit digit of \( 3355 \) is \( 5 \).

### Final Answer:
The unit digit of \( \lambda \) is \( 5 \).