Without expanding, show that the value of each of the following determ

1.	Without expanding, show that the value of each of the following determinants is zero :\(\begin{vmatrix}8 &2 & 7 \\[0.3em]12 & 3 & 5 \\[0.3em]16 &4 & 3\end{vmatrix}\)
Answer» Let Δ = \(\begin{vmatrix}8 &2 & 7 \\[0.3em]12 & 3 & 5 \\[0.3em]16 &4 & 3\end{vmatrix}\) Applying R₃→ R₃ – R₂, we get ⇒ Δ = \(\begin{vmatrix}8 &2 & 7 \\[0.3em]12 & 3 & 5 \\[0.3em]4 &1 & -2\end{vmatrix}\) Applying R₂ → R₂ – R₁, we get ⇒ Δ = \(\begin{vmatrix}8 &2 & 7 \\[0.3em]4 & 1 & -2 \\[0.3em]4 &1 & -2\end{vmatrix}\) As, R₂ = R₃, Therefore the value of the determinant is zero.

Without expanding, show that the value of each of the following determinants is zero :\(\begin{vmatrix}8 &2 & 7 \\[0.3em]12 & 3 & 5 \\[0.3em]16 &4 & 3\end{vmatrix}\)

Answer»

Let Δ = \(\begin{vmatrix}8 &2 & 7 \\[0.3em]12 & 3 & 5 \\[0.3em]16 &4 & 3\end{vmatrix}\)

Applying R₃→ R₃ – R₂, we get

⇒ Δ = \(\begin{vmatrix}8 &2 & 7 \\[0.3em]12 & 3 & 5 \\[0.3em]4 &1 & -2\end{vmatrix}\)

Applying R₂ → R₂ – R₁, we get

⇒ Δ = \(\begin{vmatrix}8 &2 & 7 \\[0.3em]4 & 1 & -2 \\[0.3em]4 &1 & -2\end{vmatrix}\)

As,

R₂ = R₃,

Therefore the value of the determinant is zero.