If A is a square matrix of order `2xx2` such that `|A|=27`, then sum of the infinite series `|A|+|1/2A|+|1/4 A|+|1/8 A|+...` is equal to _______ .

Explanation

To solve the problem, we need to find the sum of the infinite series given that \( |A| = 27 \).

### Step-by-Step Solution:

1. **Understanding the Determinant of Scaled Matrices**:
The determinant of a scaled matrix can be expressed as:
\[ |kA| = k^n |A| \]
where \( n \) is the order of the matrix. For a \( 2 \times 2 \) matrix, \( n = 2 \). Therefore:
\[ |kA| = k^2 |A| \]

2. **Calculating Determinants for Each Term**:
- For \( |A| \):
\[ |A| = 27 \]
- For \( |(1/2)A| \):
\[ |(1/2)A| = \left(\frac{1}{2}\right)^2 |A| = \frac{1}{4} |A| = \frac{1}{4} \times 27 = \frac{27}{4} \]
- For \( |(1/4)A| \):
\[ |(1/4)A| = \left(\frac{1}{4}\right)^2 |A| = \frac{1}{16} |A| = \frac{1}{16} \times 27 = \frac{27}{16} \]
- For \( |(1/8)A| \):
\[ |(1/8)A| = \left(\frac{1}{8}\right)^2 |A| = \frac{1}{64} |A| = \frac{1}{64} \times 27 = \frac{27}{64} \]

3. **Identifying the Infinite Series**:
The series we need to sum is:
\[ |A| + |(1/2)A| + |(1/4)A| + |(1/8)A| + \ldots \]
This can be rewritten as:
\[ 27 + \frac{27}{4} + \frac{27}{16} + \frac{27}{64} + \ldots \]
Factoring out \( 27 \):
\[ 27 \left( 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \ldots \right) \]

4. **Recognizing the Geometric Series**:
The series inside the parentheses is a geometric series with:
- First term \( a = 1 \)
- Common ratio \( r = \frac{1}{4} \)

5. **Sum of the Infinite Geometric Series**:
The sum \( S \) of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{1}{1 - \frac{1}{4}} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \]

6. **Calculating the Final Sum**:
Now substituting back into our expression:
\[ \text{Total Sum} = 27 \times \frac{4}{3} = 36 \]

Thus, the sum of the infinite series is:
\[ \boxed{36} \]