Without expanding, prove that `|a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|`

Explanation

To prove that the determinants \( |a \, b \, c \, x \, y \, z \, p \, q \, r| = |x \, y \, z \, p \, q \, r \, a \, b \, c| = |y \, b \, q \, x \, a \, p \, z \, c \, r| \) without expanding, we will use properties of determinants, specifically row and column transformations.

### Step-by-step Solution:

1. **Define the Determinants**:
Let:
\[
A = |a \, b \, c \, x \, y \, z \, p \, q \, r|
...