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 can use properties of determinants. Here’s a step-by-step solution:

### Step 1: Define the Determinants
Let:
- \( A = |a \, b \, c \, x \, y \, z \, p \, q \, r| \)
- \( B = |x \, y \, z \, p \, q \, r \, a \, b \, c| \)
- \( C = |y \, b \, q \, x \, a \, p \, z \, c \, r| \)

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