First slide

Introduction to Determinants

Question

If $Δ = |\begin{array}{ccc} 1 + y & 1 - y & 1 - y \\ 1 - y & 1 + y & 1 - y \\ 1 - y & 1 - y & 1 + y \end{array}| = 0$ , then value of $y$ are

Moderate

A

0, 3
B

2, – 1
C

– 1, 3
D

0, 2

Solution

Using $C_{1} \to C_{1} + C_{2} + C_{3}$ , we get
$Δ = (3 - y) |\begin{array}{lll} 1 1 - y 1 - y \\ 1 1 + y 1 - y \\ 1 1 - y 1 + y \end{array}|$
Applying $R_{3} \to R_{3} - R_{2}, R_{2} \to R_{2} - R_{1}$ we get
$\begin{array}{l} Δ = (3 - y) |\begin{array}{ccc} 1 & 1 - y & 1 - y \\ 0 & 2 y & 0 \\ 0 & - 2 y & 2 y \end{array}| = (3 - y) (4 y^{2}) \\ ∴ Δ = 0 \Rightarrow y = 0 or y = 3 . \end{array}$

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