Question

The value of x so that the matrix $\begin{bmatrix} x + a & b & c \\ a & x + b & c \\ a & b & x + c \end{bmatrix}$has rank 3 is

• A. $x \neq 0$
• B. $x = a + b + c$
• C. $x \neq 0$and $x \neq - (a + b + c)$
• D. $x = 0,x = a + b + c$

Answer 1

Answer: (C)

$x \neq 0$and $x \neq - (a + b + c)$

Answer 2

Since rank is 3, $|A|_{3 \times 3} \neq 0$, $\left| \begin{matrix} x + a & b & c \\ a & x + b & c \\ a & b & x + c \end{matrix} \right|_{3 \times 3} \neq 0$

$\left| \begin{matrix} x + a + b + c & b & c \\ x + a + b + c & x + b & c \\ x + a + b + c & b & x + c \end{matrix} \right| \neq 0$, Applying

$(C_{1} \rightarrow C_{1} + C_{2} + C_{3})$

$(x + a + b + c)\left| \begin{matrix} 1 & b & c \\ 1 & x + b & c \\ 1 & b & x + c \end{matrix} \right| \neq 0$

⇒$x + a + b + c \neq 0,\left| \begin{matrix} 1 & b & c \\ 1 & x + b & c \\ 1 & b & x + c \end{matrix} \right| \neq 0$

$x \neq - (a + b + c)$, $\left| \begin{matrix} 0 & - x & c \\ 0 & x & - x \\ 1 & b & x + c \end{matrix} \right| \neq 0$ ⇒ $x \neq 0$