If a₁, a₂, a₃, ..., a₉ are in G.P., then what is the value of the determinant ?

Concept:

Geometric Progression:

• The ratio between any two consecutive terms of a geometric progression is a fixed constant, called the common ratio (r) of the progression.

Logarithms:

• log a - log b = .

Determinants:

• A linear combination of the rows/columns does not affect the value of the determinant.
• If two rows/columns of a given matrix are interchanged, then the value of the determinant gets multiplied by -1.
• If a row/column of a given matrix is multiplied by a scalar k, then the value of the determinant is also multiplied by k.

Calculation:

Let the common ratio of the given G.P. be r.

Let D = .

Using C₁ → C₁ - C₂ and C2 → C2 - C3, we get:

⇒ D = 

⇒ D = 

⇒ D = 

Using C₁ → C₁ - C₂, we get:

⇒ D = 

Expanding along C₁, we get:

⇒ D = 0.