Question

Let x and y be two vectors in a 3 dimensional space and <x, y> denote their dot product. Then the determinant
det$\left[\begin{array}{cc}<x.x>& <x.y>\\ <y.x>& <y.y>\end{array}\right]$

• A. is zero when x and y are linearly independent.
• B. is positive when x and y are linearly independent.
• C. is non-zero for all non zero x and y
• D. is zero only when either x and y is zero.

Answer 1

The correct option is B is positive when x and y are linearly independent.

For the sake of simplicity we wil slove this question in 2D.
Let $\Delta =\left[\begin{array}{cc}x.x& x.y\\ y.x& y.y\end{array}\right]$
Let $x=x_1\hat{i}+x_2\hat{j}$
$y=y_1\hat{i}+y_2\hat{j}$
$x.x=x_1^2+x_2^2$
$x.y=x_1{y}_1+x_2{y}_2$
$y.y=y_1^2+y_2^2$

$|A|=\left|\begin{array}{cc}{x}_1^2+x_2^2& x_1{y}_1+x_2{y}_2\\ x_1{y}_1+x_2{y}_2& y_1^2+y_2^2\end{array}\right|$

$=(x_1^2+x_2^2)\left(y_1^2\right)\left(y_1^2\right)-(x_1{y}_1+x_2{y}_2{)}^2$
$=x_1^2{y}_1^2+x_1^2{y}_2^2+x_2^2{y}_1^2+x_2^2{y}_2^2-x_1^2{y}_1^2-x_2^2{y}_2^2-2x_1{x}_2{y}_1{y}_2$
$\Delta =(x_1{y}_2-x_2{y}_2{)}^2=${o, when x & y are LD
>0, when x %y are LI

$\Rightarrow$ For linearly independence $\Delta$ is positive