First slide

Applications of determinant

Question

The values of $' x'$ for which the matrix is $[\begin{matrix} x + a & b & c \\ a & x + b & c \\ a & b & x + c \end{matrix}]$ non-singular are

Difficult

A

$R - {0}$
B

$R - {- (a + b + c)}$
C

$R - {0, - (a + b + c)}$
D

none of these

Solution

$|\begin{matrix} x + a & b & c \\ a & x + b & c \\ a & b & x + c \end{matrix}|$
$C_{1} \to C_{1} + C_{2} + C_{3}$
$(x + a + b + c) |\begin{matrix} 1 & b & c \\ 1 & x + b & c \\ 1 & b & x + c \end{matrix}|$
$\begin{matrix} R_{2} \to R_{2} - R_{1} \\ R_{3} \to R_{3} - R_{1} \end{matrix}$
$(x + a + b + c) |\begin{matrix} 1 & b & c \\ 0 & x & 0 \\ 0 & 0 & x \end{matrix}| = x^{2} (x + a + b + c)$
Given matrix is non singular
$\Rightarrow x^{2} (x + a + b + c) \neq 0 \Rightarrow x \neq 0, x \neq - a - b - c$

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