First slide

Introduction to Determinants

Question

If $Δ = |\begin{matrix} 3 & 4 & 5 & x \\ 4 & 5 & 6 & y \\ 5 & 6 & 7 & z \\ x & y & z & 0 \end{matrix}| = 0$ , then

Moderate

A

x, y, z are in A.P.
B

x, y, z are in G.P.
C

x, y, z are in H.P.
D

none of these

Solution

Applying $R_{1} \to R_{1} + R_{3} - 2 R_{2}$ , we get
$\begin{matrix} Δ = |\begin{array}{cccc} 0 & 0 & 0 & x + z - 2 y \\ 4 & 5 & 6 & y \\ 5 & 6 & 7 & z \\ x & y & z & 0 \end{array}| \\ = - (x + z - 2 y) |\begin{matrix} 4 & 5 & 6 \\ 5 & 6 & 7 \\ x & y & z \end{matrix}| [Expanding along R_{1}] \\ = - (x + z - 2 y) |\begin{array}{ccc} 0 & - 1 & 6 \\ 0 & - 1 & 7 \\ x - 2 y + z & y - z & z \end{array}| [Applying C_{1} \to C_{1} + C_{3} - 2 C_{2} and C_{2} \to C_{2} - C_{3}] \\ = - (x + z - 2 y)^{2} |\begin{matrix} - 1 & 6 \\ - 1 & 7 \end{matrix}| \\ = (x - 2 y + z)^{2} \end{matrix}$
Hence, $Δ = 0 \Rightarrow x, y, z$ are in A.P.

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