A and B are two non-singular square matrices of each `3xx3` such that AB = A and BA = B and `|A+B| ne 0` then

Options

A. `|A+B|=0`
B. `|A+B|=8`
C. `|A-B|=1`
D. `|A+B|=2`

Answer

To solve the problem step by step, we need to analyze the given conditions and derive the necessary conclusions.

### Step 1: Understand the Given Conditions
We have two non-singular matrices \( A \) and \( B \) of size \( 3 \times 3 \) such that:
1. \( AB = A \)
2. \( BA = B \)
3. \( |A + B| \neq 0 \)

### Step 2: Use the First Condition \( AB = A \)
From the equation \( AB = A \), we can rearrange it as:
\[ AB - A = 0 \implies A(B - I) = 0 \]
Since \( A \) is non-singular (invertible), we can conclude that:
\[ B - I = 0 \implies B = I \]

### Step 3: Use the Second Condition \( BA = B \)
Now substituting \( B = I \) into the second condition \( BA = B \):
\[ IA = I \implies A = I \]

### Step 4: Calculate \( A + B \)
Now that we have \( A = I \) and \( B = I \):
\[ A + B = I + I = 2I \]

### Step 5: Calculate the Determinant
Next, we need to find the determinant of \( A + B \):
\[ |A + B| = |2I| \]
The determinant of a scalar multiple of the identity matrix is given by:
\[ |kI| = k^n \quad \text{(where \( n \) is the size of the matrix)} \]
In our case, \( k = 2 \) and \( n = 3 \):
\[ |2I| = 2^3 = 8 \]

### Step 6: Conclusion
Since \( |A + B| = 8 \) and \( 8 \neq 0 \), the condition \( |A + B| \neq 0 \) is satisfied.

### Final Answer
Thus, we conclude that the conditions hold true and the determinant of \( A + B \) is \( 8 \).