If `A` is an idempotent matrix satisfying, `(I-0. 4 A)^(-1)=I-alphaA ,w h e r eI` is the unit matrix of the same order as that of `A ,` then th value of `|9alpha|` is equal to ________.

Explanation

To solve the problem, we need to find the value of \( |9\alpha| \) given that \( A \) is an idempotent matrix satisfying the equation:

\[ (I - 0.4A)^{-1} = I - \alpha A \]

### Step 1: Understand the properties of the idempotent matrix
Since \( A \) is idempotent, we have:

\[ A^2 = A \]

### Step 2: Multiply both sides by \( (I - 0.4A) \)
We can multiply both sides of the given equation by \( (I - 0.4A) \):

\[ (I - 0.4A)(I - 0.4A)^{-1} = (I - 0.4A)(I - \alpha A) \]

This simplifies to:

\[ I = (I - 0.4A)(I - \alpha A) \]

### Step 3: Expand the right-hand side
Now, we expand the right-hand side:

\[ I = I - \alpha A - 0.4A + 0.4\alpha A^2 \]

### Step 4: Substitute \( A^2 \) with \( A \)
Since \( A \) is idempotent, we can replace \( A^2 \) with \( A \):

\[ I = I - \alpha A - 0.4A + 0.4\alpha A \]

### Step 5: Combine like terms
Now, we can combine the terms involving \( A \):

\[ I = I - (\alpha + 0.4 - 0.4\alpha)A \]

### Step 6: Set the coefficients of \( A \) to zero
Since the identity matrix \( I \) must hold for all \( A \), we set the coefficient of \( A \) to zero:

\[ \alpha + 0.4 - 0.4\alpha = 0 \]

### Step 7: Solve for \( \alpha \)
Rearranging the equation gives:

\[ \alpha - 0.4\alpha + 0.4 = 0 \]

This simplifies to:

\[ (1 - 0.4)\alpha = -0.4 \]

\[ 0.6\alpha = -0.4 \]

\[ \alpha = \frac{-0.4}{0.6} = \frac{-2}{3} \]

### Step 8: Calculate \( |9\alpha| \)
Now, we need to find \( |9\alpha| \):

\[ |9\alpha| = |9 \times \frac{-2}{3}| = |-6| = 6 \]

Thus, the final answer is:

\[ \boxed{6} \]