Question

If $ab + bc + ca = 0$ and$\left| \begin{matrix} a - x & c & b \\ c & b - x & a \\ b & a & c - x \end{matrix} \right| = 0$, then one of the value of x is.

• A. $(a^{2} + b^{2} + c^{2})^{\frac{1}{2}}$
• B. $\left\lbrack \frac{3}{2}(a^{2} + b^{2} + c^{2}) \right\rbrack^{\frac{1}{2}}$
• C. $\left\lbrack \frac{1}{2}(a^{2} + b^{2} + c^{2}) \right\rbrack^{\frac{1}{2}}$
• D. None of these

Answer 1

Answer: (A)

$(a^{2} + b^{2} + c^{2})^{\frac{1}{2}}$

Answer 2

Applying $A^{- 1} = \frac{1}{- 2}\begin{bmatrix}

• 1 & 1 & - 1 \ 8 & - 6 & 2 \
• 5 & 3 & - 1 \end{bmatrix}$

∴$A_{32} = 2$

⇒ $( a + b + c - x ) \left[ \left\{ ( b - x ) ( c - x ) - a ^ { 2 } \right\} + c ( a - c + x ) \right\}$ $A_{12} = 8,$

⇒$|A| = - 2 \neq 0A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{bmatrix}$

⇒ $x = 0$

1 \times 1 + 2 \times 2 + ( - 1)(0) \\ 3 \times 1 + 0 \times 2 + 2 \times 0 \\ 4 \times 1 + 5 \times 2 + 0 \times 0 \end{matrix} \right.\ $$ ∴ $= \begin{bmatrix} 1 & - 1 & 0 \\ - 2 & 3 & - 4 \\ - 2 & 3 & - 3 \end{bmatrix}$ and $A = \begin{bmatrix} 1 & 2 & - 1 \\ 3 & 0 & 2 \\ 4 & 5 & 0 \end{bmatrix}$.