Question

Consider a 3 x 3 real symmetric matrix S such that two of its eigen values are $a\ne 0,b\ne 0$ with respective eigen vectors $\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right],\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]$. If $a\ne b$ then $x_1{y}_1+x_2{y}_2+x_3{y}_3$ equals

• A. a
• B. b
• C. ab
• D. 0

Answer 1

The correct option is D 0

The eigen vectors of a real symmetric matrix corresponding to distinct eigen values are othogonal.

so, $X.Y=0\Rightarrow \left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]=x_1{y}_1+x_2{y}_2+x_3{y}_3=0$