Evaluate the following determinant :\(\begin{vmatrix} 67& 19& 21 \\[0.

1.	Evaluate the following determinant :\(\begin{vmatrix} 67& 19& 21 \\[0.3em] 39& 13 &14 \\[0.3em] 81 &24 & 26 \end{vmatrix}\)
Answer» Let Δ = \(\begin{vmatrix} 67& 19& 21 \\[0.3em] 39& 13 &14 \\[0.3em] 81 &24 & 26 \end{vmatrix}\) Applying, C₁ → C₁ – 4 C₃, we get, ⇒ Δ = \(\begin{vmatrix} 4& 19& 21 \\[0.3em] -3& 13 &14 \\[0.3em] -3&24 & 26 \end{vmatrix}\) Applying, R₁ → R₁ + R₂ and R₃→ R₃ – R₂, we get ⇒ Δ = \(\begin{vmatrix} 1& 32& 35 \\[0.3em] -3& 13 &14 \\[0.3em] 0&11 & 12 \end{vmatrix}\) Now, applying R₂→ R₂ + 3R₁, we get, ⇒ Δ = \(\begin{vmatrix} 1& 32& 35 \\[0.3em] 0& 109 &119 \\[0.3em] 0&11 & 12 \end{vmatrix}\) = 1[(109)(12) – (119)(11)] = 1308 – 1309 = – 1 So, Δ = – 1

Evaluate the following determinant :\(\begin{vmatrix} 67& 19& 21 \\[0.3em] 39& 13 &14 \\[0.3em] 81 &24 & 26 \end{vmatrix}\)

Answer»

Let Δ = \(\begin{vmatrix} 67& 19& 21 \\[0.3em] 39& 13 &14 \\[0.3em] 81 &24 & 26 \end{vmatrix}\)

Applying, C₁ → C₁ – 4 C₃, we get,

⇒ Δ = \(\begin{vmatrix} 4& 19& 21 \\[0.3em] -3& 13 &14 \\[0.3em] -3&24 & 26 \end{vmatrix}\)

Applying, R₁ → R₁ + R₂ and R₃→ R₃ – R₂, we get

⇒ Δ = \(\begin{vmatrix} 1& 32& 35 \\[0.3em] -3& 13 &14 \\[0.3em] 0&11 & 12 \end{vmatrix}\)

Now, applying R₂→ R₂ + 3R₁, we get,

⇒ Δ = \(\begin{vmatrix} 1& 32& 35 \\[0.3em] 0& 109 &119 \\[0.3em] 0&11 & 12 \end{vmatrix}\)

= 1[(109)(12) – (119)(11)]

= 1308 – 1309

= – 1

So,

Δ = – 1