If `A = |(a,b,c),(x,y,z),(p,q,r)| and B = |(q,-b,y),(-p,a,-x),(r,-c,z)|`, then

Options

A. `A = 2B`
B. `A = B`
C. `A = -B`
D. none of these

Answer

To solve the problem, we need to analyze the determinants \( A \) and \( B \) given in the question. We will manipulate the determinants step by step to find a relationship between them.

Given:
\[ A = \begin{vmatrix} a & b & c \\ x & y & z \\ p & q & r \end{vmatrix} \]
\[ B = \begin{vmatrix} q & -b & y \\ -p & a & -x \\ r & -c & z \end{vmatrix} \]

### Step 1: Interchange Columns
We can interchange columns in the determinant \( B \). Let's interchange column 1 and column 2.

When we interchange two columns, we introduce a negative sign:
\[ B = -\begin{vmatrix} -b & q & y \\ a & -p & -x \\ -c & r & z \end{vmatrix} \]

**Hint:** Remember that interchanging two columns in a determinant changes the sign of the determinant.

### Step 2: Interchange Rows
Next, we can interchange row 1 and row 2 in the determinant we obtained from the previous step. This will again introduce a negative sign:
\[ B = -\begin{vmatrix} a & -p & -x \\ -b & q & y \\ -c & r & z \end{vmatrix} \]
\[ B = -(-1)\begin{vmatrix} -b & q & y \\ a & -p & -x \\ -c & r & z \end{vmatrix} = \begin{vmatrix} -b & q & y \\ a & -p & -x \\ -c & r & z \end{vmatrix} \]

**Hint:** Interchanging rows also changes the sign of the determinant.

### Step 3: Factor Out Constants
Now, we can factor out the negative signs from the first column and the second row:
\[ B = (-1)(-1)\begin{vmatrix} b & q & y \\ a & p & x \\ c & r & z \end{vmatrix} = \begin{vmatrix} b & q & y \\ a & p & x \\ c & r & z \end{vmatrix} \]

**Hint:** When you factor out a negative sign from a column or row, it changes the sign of the determinant.

### Step 4: Rearranging the Determinant
Now we can rearrange the determinant \( B \) to match the structure of \( A \):
\[ B = \begin{vmatrix} a & b & c \\ x & y & z \\ p & q & r \end{vmatrix} = A \]

### Step 5: Final Relationship
Thus, we find that:
\[ B = -A \]

### Conclusion
The relationship between the determinants is:
\[ \text{det}(B) = -\text{det}(A) \]

The correct option is that \( B \) is equal to the negative of \( A \).