`A=[[2,1],[4,1]]; B=[[3,4],[2,3]]` & `c=[[3,-4],[-2,3]]`, `tr(A)+tr[(ABC)/2]+tr[(A(BC)^2)/4]+tr[(A(BC)^2)/8]+.....oo` is:

Options

A. 6
B. 9
C. 12
D. none of these

Explanation

To solve the problem step by step, we will follow the instructions outlined in the video transcript.

### Step 1: Define the matrices
We have the following matrices:
- \( A = \begin{bmatrix} 2 & 1 \\ 4 & 1 \end{bmatrix} \)
- \( B = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix} \)
- \( C = \begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix} \)

### Step 2: Calculate the product \( BC \)
To find \( BC \):
\[ BC = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix} \]
Calculating the elements:
- First row, first column: \( 3 \cdot 3 + 4 \cdot (-2) = 9 - 8 = 1 \)
- First row, second column: \( 3 \cdot (-4) + 4 \cdot 3 = -12 + 12 = 0 \)
- Second row, first column: \( 2 \cdot 3 + 3 \cdot (-2) = 6 - 6 = 0 \)
- Second row, second column: \( 2 \cdot (-4) + 3 \cdot 3 = -8 + 9 = 1 \)

Thus, we have:
\[ BC = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I \]
where \( I \) is the identity matrix.

### Step 3: Analyze the series
The series we need to evaluate is:
\[ \text{tr}(A) + \text{tr}\left(\frac{ABC}{2}\right) + \text{tr}\left(\frac{AB^2C^2}{4}\right) + \text{tr}\left(\frac{AB^3C^3}{8}\right) + \ldots \]

Since \( BC = I \), we can simplify:
- \( ABC = A \cdot I = A \)
- \( AB^2C^2 = A \cdot I^2 = A \)
- \( AB^3C^3 = A \cdot I^3 = A \)

Thus, the series simplifies to:
\[ \text{tr}(A) + \text{tr}\left(\frac{A}{2}\right) + \text{tr}\left(\frac{A}{4}\right) + \text{tr}\left(\frac{A}{8}\right) + \ldots \]

### Step 4: Factor out \( \text{tr}(A) \)
Let \( x = \text{tr}(A) \). The series becomes:
\[ x + \frac{x}{2} + \frac{x}{4} + \frac{x}{8} + \ldots \]

### Step 5: Sum the infinite geometric series
The series is a geometric series with:
- First term \( a = x \)
- Common ratio \( r = \frac{1}{2} \)

The sum \( S \) of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} = \frac{x}{1 - \frac{1}{2}} = \frac{x}{\frac{1}{2}} = 2x \]

### Step 6: Calculate \( \text{tr}(A) \)
To find \( \text{tr}(A) \):
\[ \text{tr}(A) = 2 + 1 = 3 \]

### Step 7: Final result
Substituting back, we have:
\[ S = 2 \cdot \text{tr}(A) = 2 \cdot 3 = 6 \]

Thus, the final answer is:
\[ \boxed{6} \]