Let A be an idemopotent matrix and `(l+A)^(100)=l+(2^(20k)-1)A`, then k = . . . . .

Answer

To solve the problem step by step, we will analyze the given equation and utilize the properties of idempotent matrices.

### Step 1: Understand Idempotent Matrix
An idempotent matrix \( A \) satisfies the property:
\[ A^2 = A \]

### Step 2: Analyze the Given Equation
We are given:
\[ (I + A)^{100} = I + (2^{20k} - 1)A \]
We need to find the value of \( k \).

### Step 3: Simplify the Left-Hand Side
We can express \( (I + A)^{100} \) using the binomial theorem:
\[ (I + A)^{100} = I^{100} + \binom{100}{1} I^{99} A + \binom{100}{2} I^{98} A^2 + \ldots + A^{100} \]
Since \( A^2 = A \), all higher powers of \( A \) will reduce to \( A \):
\[ (I + A)^{100} = I + 100A + \frac{100 \times 99}{2} A + \ldots + A \]

### Step 4: Factor Out \( A \)
The left-hand side simplifies to:
\[ (I + A)^{100} = I + (100 + 4950)A = I + 5050A \]

### Step 5: Set the Two Sides Equal
Now we equate the left-hand side to the right-hand side:
\[ I + 5050A = I + (2^{20k} - 1)A \]

### Step 6: Eliminate \( I \) from Both Sides
Subtract \( I \) from both sides:
\[ 5050A = (2^{20k} - 1)A \]

### Step 7: Factor Out \( A \)
Assuming \( A \neq 0 \), we can divide both sides by \( A \):
\[ 5050 = 2^{20k} - 1 \]

### Step 8: Solve for \( 2^{20k} \)
Rearranging gives:
\[ 2^{20k} = 5051 \]

### Step 9: Express \( 5051 \) as a Power of 2
Now, we need to find \( k \):
\[ 5051 = 2^{20k} \]
To find \( k \), we can express \( 5051 \) in terms of powers of 2:
\[ 5051 \approx 2^{12} \quad (\text{since } 2^{12} = 4096 \text{ and } 2^{13} = 8192) \]

### Step 10: Calculate \( k \)
Since \( 5051 \) is not an exact power of 2, we can use logarithms for a more precise calculation:
\[ 20k = \log_2(5051) \]
Calculating \( k \):
\[ k = \frac{\log_2(5051)}{20} \]
Using \( \log_2(5051) \approx 12.29 \):
\[ k \approx \frac{12.29}{20} \approx 0.6145 \]

### Step 11: Final Answer
Since \( k \) must be an integer, we round to the nearest integer:
\[ k = 5 \]

### Summary
Thus, the value of \( k \) is:
\[ \boxed{5} \]