Let `P_(n)` be s square matrix of order 3 such that `P_(n)=[a_(ij)]`, where `a_(ij)=(3i+j)/(4^(n))` for `1leile3, 1lejle3`. Then the value of `lim_(nrarroo)T_(r)(4P_(1)+4^(2)P_(2)……….4^(n)P_(n))` is (where `T_(r)(A)` denotes trace of matrix A i.e sum of principle diagonal elements of A)

Options

A. 7
B. 8
C. `(25)/(3)`
D. 9

Answer

To solve the problem step by step, we will analyze the matrix \( P_n \) and compute the limit of the trace expression given in the question.

### Step 1: Define the Matrix \( P_n \)
The elements of the matrix \( P_n \) are given by:
\[ a_{ij} = \frac{3i + j}{4^n} \]
for \( i, j = 1, 2, 3 \). Thus, the matrix \( P_n \) can be written as:
\[ P_n = \begin{bmatrix} \frac{4}{4^n} & \frac{5}{4^n} & \frac{6}{4^n} \\ \frac{6}{4^n} & \frac{7}{4^n} & \frac{8}{4^n} \\ \frac{9}{4^n} & \frac{10}{4^n} & \frac{11}{4^n} \end{bmatrix} \]

### Step 2: Calculate the Trace of \( P_n \)
The trace of a matrix is the sum of its diagonal elements. Therefore, we compute:
\[ \text{Trace}(P_n) = a_{11} + a_{22} + a_{33} \]
Calculating each term:
- \( a_{11} = \frac{4}{4^n} \)
- \( a_{22} = \frac{8}{4^n} \)
- \( a_{33} = \frac{12}{4^n} \)

Thus, the trace becomes:
\[ \text{Trace}(P_n) = \frac{4}{4^n} + \frac{8}{4^n} + \frac{12}{4^n} = \frac{24}{4^n} \]

### Step 3: Compute the Expression \( T_r(4P_1 + 4^2P_2 + \ldots + 4^nP_n) \)
We need to evaluate:
\[ T_r(4P_1 + 4^2P_2 + \ldots + 4^nP_n) \]
Using the trace property \( T_r(aA) = aT_r(A) \), we have:
\[ T_r(4P_k) = 4 \cdot \text{Trace}(P_k) \]
Thus:
\[ T_r(4P_1 + 4^2P_2 + \ldots + 4^nP_n) = 4 \cdot \text{Trace}(P_1) + 4^2 \cdot \text{Trace}(P_2) + \ldots + 4^n \cdot \text{Trace}(P_n) \]
Substituting the trace values:
\[ = 4 \cdot \frac{24}{4^1} + 4^2 \cdot \frac{24}{4^2} + \ldots + 4^n \cdot \frac{24}{4^n} \]
This simplifies to:
\[ = 24(1 + 1 + \ldots + 1) = 24n \]

### Step 4: Take the Limit as \( n \to \infty \)
Now, we need to evaluate:
\[ \lim_{n \to \infty} T_r(4P_1 + 4^2P_2 + \ldots + 4^nP_n) \]
Since the expression simplifies to \( 24n \), as \( n \to \infty \), this diverges to infinity.

### Conclusion
Thus, the limit does not converge to a finite value. The value of the limit is:
\[ \lim_{n \to \infty} T_r(4P_1 + 4^2P_2 + \ldots + 4^nP_n) = \infty \]