Consider three square matrices A, B and C of order 3 such that `A^(T)=A-2B and B^(T)=B-4C`, then the incorrect option is

Options

A. `|A|=0`
B. `|B|=0`
C. `|C|=0`
D. `B=2C`

Answer

To solve the problem, we need to analyze the given equations involving the matrices A, B, and C. Let's break it down step by step.

### Step 1: Analyze the first equation
We start with the equation given in the problem:
\[ A^T = A - 2B \]

Rearranging this gives:
\[ A - A^T = 2B \]

This implies that \( B \) can be expressed as:
\[ B = \frac{1}{2}(A - A^T) \]

### Step 2: Identify properties of B
Since \( A - A^T \) is the difference of a matrix and its transpose, it is skew-symmetric. Therefore, \( B \) is also skew-symmetric because it is a scalar multiple of a skew-symmetric matrix.

### Step 3: Analyze the second equation
Now, we look at the second equation:
\[ B^T = B - 4C \]

Rearranging this gives:
\[ B - B^T = 4C \]

This implies that \( C \) can be expressed as:
\[ C = \frac{1}{4}(B - B^T) \]

### Step 4: Identify properties of C
Similarly, since \( B - B^T \) is also the difference of a matrix and its transpose, it is skew-symmetric. Therefore, \( C \) is skew-symmetric as well.

### Step 5: Determine the determinants
For any skew-symmetric matrix of odd order (like our 3x3 matrices), the determinant is zero. Thus:
\[ \text{det}(B) = 0 \]
\[ \text{det}(C) = 0 \]

### Step 6: Relate A, B, and C
Next, we take the transpose of the first equation:
\[ (A^T)^T = (A - 2B)^T \]
This simplifies to:
\[ A = A^T - 2B^T \]

Substituting \( B^T \) from the second equation:
\[ A = A^T - 2(B - 4C) \]
This simplifies to:
\[ A = A^T - 2B + 8C \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ A - A^T + 2B - 8C = 0 \]

### Step 8: Substitute for B
From our earlier result, we know:
\[ B = \frac{1}{2}(A - A^T) \]
Substituting this into the equation gives:
\[ A - A^T + (A - A^T) - 8C = 0 \]
This simplifies to:
\[ 2(A - A^T) - 8C = 0 \]
Thus:
\[ A - A^T = 4C \]

### Step 9: Express B in terms of C
From the earlier relationship \( B = 2C \), we can conclude that:
\[ B = 2C \]

### Conclusion
Now, we can check the options provided in the question to identify the incorrect one. Based on our findings, we have established relationships between A, B, and C, particularly that B is twice C.