First slide

Introduction to Determinants

Question

Let $P (x) = |\begin{array}{ccc} x & - 3 + 4 i & 3 - 4 i \\ x & - 7 i & 5 + 6 i \\ - x & 7 - 2 i & - 7 - 2 i \end{array}|$ .The number of values of x for which $P (x) = 0$ is

Moderate

A

0
B

1
C

2
D

3

Solution

Using $R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} + R_{1},$
We get
$P (x) = |\begin{array}{ccc} x & - 3 + 4 i & 3 - 4 i \\ 0 & 3 - 11 i & 2 + 10 i \\ 0 & 4 + 2 i & - 4 - 6 i \end{array}|$
As $P (x)$ is a linear polynomial, $P (x) = 0$ for exactly one value of $x .$

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