Solve the following determinant equations :|(3x-8,3,3)(3,3x-8,3)(3,3,3x-8)|= 0 ​

Question

Solve the following determinant equations : 

\(\begin{vmatrix} 3x-8 & 3 & 3 \\[0.3em] 3 & 3x-8 & 3 \\[0.3em] 3 & 3 & 3x-8 \end{vmatrix}\) = 0

Answer 1

\(\begin{vmatrix} 3x-8 & 3 & 3 \\[0.3em] 3 & 3x-8 & 3 \\[0.3em] 3 & 3 & 3x-8 \end{vmatrix}\) = 0

Let Δ = \(\begin{vmatrix} 3x-8 & 3 & 3 \\[0.3em] 3 & 3x-8 & 3 \\[0.3em] 3 & 3 & 3x-8 \end{vmatrix}\) 

We need to find the roots of Δ = 0. 

Recall that the value of a determinant remains same if we apply the operation R_i→ R_i + kR_j or C_i→ C_i + kC_j. 

Applying C₁→ C₁ + C₂, we get

Expanding the determinant along C₁, we have

Δ = (3x – 2)(1)[(3x – 11)(3x – 11) – (0)(0)]

⇒ Δ = (3x – 2)(3x – 11)(3x – 11)

∴ Δ = (3x – 2)(3x – 11)²

The given equation is Δ = 0.

⇒ (3x – 2)(3x – 11)² = 0

Case – I : 

3x – 2 = 0 

⇒ 3x = 2

∴ x = \(\frac{2}{3}\)

Case – II :

(3x – 11)² = 0 

⇒ 3x – 11 = 0 

⇒ 3x = 11

 ∴ x = \(\frac{11}{3}\)

Thus, 

\(\frac{2}{3}\) and \(\frac{11}{3}\) are the roots of the given determinant equation.