If `A=[(a,b,c),(x,y,z),(p,q,r)], B=[(q,-b,y),(-p,a,-x),(r,-c,z)]` and If A is invertible, then which of the following is not true ?

Options

A. `|A|=|B|`
B. `|A|=-|B|`
C. `|"adj A"|=|"adj B"|`
D. A is invertible if and only if B is invertible

Explanation

To solve the problem, we need to analyze the matrices A and B, and determine which statement about them is not true given that A is invertible.

### Step 1: Write down the matrices A and B
We have:
\[ A = \begin{pmatrix} a & b & c \\ x & y & z \\ p & q & r \end{pmatrix} \]
\[ B = \begin{pmatrix} q & -b & y \\ -p & a & -x \\ r & -c & z \end{pmatrix} \]

### Step 2: Check the properties of A
Since A is given to be invertible, it means that its determinant is non-zero, i.e., \( \text{det}(A) \neq 0 \).

### Step 3: Analyze the matrix B
To understand the relationship between A and B, we can perform row operations on B to see if we can express it in terms of A.

### Step 4: Perform row operations on B
1. **Interchange R1 and R2**:
\[ B = \begin{pmatrix} -p & a & -x \\ q & -b & y \\ r & -c & z \end{pmatrix} \]

2. **Multiply R1 by -1**:
\[ B = \begin{pmatrix} p & -a & x \\ q & -b & y \\ r & -c & z \end{pmatrix} \]

### Step 5: Compare B with A
Now we can see that the first row of B has been transformed, and we need to check if we can express B in terms of A.

### Step 6: Establish a relationship
If we take the negative of the first row of A and rearrange the rows, we can see that:
\[ B = \begin{pmatrix} p & -a & x \\ q & -b & y \\ r & -c & z \end{pmatrix} \]
This suggests that B can be related to A, but we need to check the specific statements given in the options to determine which one is not true.

### Conclusion
Without the specific options provided in the question, we cannot definitively conclude which statement is not true. However, we have established that B can be derived from A through row operations, and since A is invertible, it implies that B has certain properties as well.