`A_(1)=[a_(1)]`
`A_(2)=[{:(,a_(2),a_(3)),(,a_(4),a_(5)):}]`
`A_(3)=[{:(,a_(6),a_(7),a_(8)),(,a_(9),a_(10),a_(11)),(,a_(12),a_(13),a_(14)):}]......A_(n)=[.......]`
Where, `a_(r)=[log_(2)r]([.])` denotes greatest integer). Then trace of `A_(10)`

Answer

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

### Step 1: Understand the structure of the matrices
The matrix \( A_n \) is an \( n \times n \) matrix. The elements of these matrices are defined by \( a_r = \lfloor \log_2 r \rfloor \), where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).

### Step 2: Determine the number of elements in \( A_{10} \)
Since \( A_{10} \) is a \( 10 \times 10 \) matrix, it contains \( 10^2 = 100 \) elements. The elements are indexed from \( A_1 \) to \( A_{100} \).

### Step 3: Identify the starting and ending indices for \( A_{10} \)
The elements in \( A_{10} \) start from \( A_{286} \) (which is \( A_1 + A_2 + A_3 + A_4 + A_5 + A_6 + A_7 + A_8 + A_9 = 285 + 1 \)) and end at \( A_{385} \).

### Step 4: Identify the diagonal elements of \( A_{10} \)
The diagonal elements of \( A_{10} \) are:
- \( A_{286} \)
- \( A_{297} \)
- \( A_{308} \)
- \( A_{319} \)
- \( A_{330} \)
- \( A_{341} \)
- \( A_{352} \)
- \( A_{363} \)
- \( A_{374} \)
- \( A_{385} \)

### Step 5: Calculate the trace of \( A_{10} \)
The trace of a matrix is the sum of its diagonal elements. Thus, we need to calculate:
\[ \text{Trace}(A_{10}) = a_{286} + a_{297} + a_{308} + a_{319} + a_{330} + a_{341} + a_{352} + a_{363} + a_{374} + a_{385} \]

### Step 6: Calculate each \( a_r \)
Using the formula \( a_r = \lfloor \log_2 r \rfloor \):
- \( a_{286} = \lfloor \log_2 286 \rfloor = 8 \)
- \( a_{297} = \lfloor \log_2 297 \rfloor = 8 \)
- \( a_{308} = \lfloor \log_2 308 \rfloor = 8 \)
- \( a_{319} = \lfloor \log_2 319 \rfloor = 8 \)
- \( a_{330} = \lfloor \log_2 330 \rfloor = 8 \)
- \( a_{341} = \lfloor \log_2 341 \rfloor = 8 \)
- \( a_{352} = \lfloor \log_2 352 \rfloor = 8 \)
- \( a_{363} = \lfloor \log_2 363 \rfloor = 8 \)
- \( a_{374} = \lfloor \log_2 374 \rfloor = 8 \)
- \( a_{385} = \lfloor \log_2 385 \rfloor = 8 \)

### Step 7: Sum the diagonal elements
Now, we sum all the calculated values:
\[ \text{Trace}(A_{10}) = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 80 \]

### Final Answer
Thus, the trace of \( A_{10} \) is \( \boxed{80} \).

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