A be a square matrix of order `2` with `|A| ne 0` such that `|A+|A|adj(A)|=0`, where `adj(A)` is a adjoint of matrix `A`, then the value of `|A-|A|adj(A)|` is

Options

A. `1`
B. `2`
C. `3`
D. `4`

Explanation

To solve the problem, we need to find the value of \(|A - |A| \cdot \text{adj}(A)|\) given the condition \(|A + |A| \cdot \text{adj}(A)| = 0\).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We start with the equation:
\[ |A + |A| \cdot \text{adj}(A)| = 0 \]
This implies that the matrix \(A + |A| \cdot \text{adj}(A)\) is singular.

2. **Using Properties of Determinants**:
Recall that for a \(2 \times 2\) matrix \(A\):
\[ \text{adj}(A) = |A| A^{-1} \]
and the determinant of the adjoint of \(A\) is given by:
\[ |\text{adj}(A)| = |A|^{n-1} \quad \text{for an } n \times n \text{ matrix} \]
For a \(2 \times 2\) matrix, this means:
\[ |\text{adj}(A)| = |A|^{1} = |A| \]

3. **Substituting into the Given Condition**:
Substitute \(|\text{adj}(A)|\) into the determinant condition:
\[ |A + |A|^2| = 0 \]

4. **Setting Up the Matrix**:
Let \(A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}\). The determinant is:
\[ |A| = mq - np \]

5. **Calculating \(|A + |A| \cdot \text{adj}(A)|\)**:
The adjoint of \(A\) is:
\[ \text{adj}(A) = \begin{pmatrix} q & -n \\ -p & m \end{pmatrix} \]
Therefore,
\[ |A| \cdot \text{adj}(A) = (mq - np) \begin{pmatrix} q & -n \\ -p & m \end{pmatrix} \]
Now, adding \(A\):
\[ A + |A| \cdot \text{adj}(A) = \begin{pmatrix} m + (mq - np)q & n - (mq - np)n \\ p - (mq - np)p & q + (mq - np)m \end{pmatrix} \]

6. **Finding the Determinant**:
Calculate the determinant of the resulting matrix:
\[ |A + |A| \cdot \text{adj}(A)| = 0 \]
This leads to a polynomial equation in terms of \(d = |A|\).

7. **Finding \(|A - |A| \cdot \text{adj}(A)|\)**:
Now we need to find:
\[ |A - |A| \cdot \text{adj}(A)| \]
Following a similar process, we compute:
\[ |A - |A| \cdot \text{adj}(A)| = |A| - |A|^2 \]

8. **Final Calculation**:
Substitute \(d = |A|\) into the expression:
\[ |A - |A| \cdot \text{adj}(A)| = d - d^2 \]
Given that \(d(d - 2) = 0\), we find:
\[ d = 0 \quad \text{or} \quad d = 2 \]
Since \(|A| \neq 0\), we conclude:
\[ d = 2 \]

9. **Final Result**:
Therefore, the value of \(|A - |A| \cdot \text{adj}(A)|\) is:
\[ \boxed{4} \]