Answer to Question #234684 in Calculus for Anuj

Question

a. Find the orthogonal and normal canonical forms of 2y^2-2yz+2zx-2xy.

b. The operation,* defined by a*b= sin(ab), is a binary operation on N

True or false with full explanation

Answer 1

$(a) \ 2y^2-2yz+2zx-2xy$

The matrix of the quadratic equation is;

$A=\begin{vmatrix} 0 & -1 &1\\ -1 & 2&0\\ 1&0 & -2 \end{vmatrix}$

The characteristic roots are;

i.e

$A=\begin{vmatrix} 0-\lambda& -1 &1\\ -1 & 2-\lambda&0\\ 1&0 & -2-\lambda \end{vmatrix}=\begin{vmatrix} 0& -1 &1\\ -1 & 2-\lambda&0\\ 1&0 & -2-\lambda \end{vmatrix}$

The characteristic vector for $\lambda = 0$ is given by;

i.e

$-1x_2 +x_3 = 0\\ -1x_1 +2x_2 = 0\\ x_1 -2x_3 = 0$

Solving for first 2,

$\dfrac{x_1}{2}= -x_2 = x_3$

giving the Eignen vector $X_1 = k_1(2,-1,1)$

When $\lambda = 3$, the characteristic vector becomes

$-4x_2 -2x_3 = 0\\ -4x_1 -1x_2 = 0\\ -2x_1 -5x_3 = 0$

which can then become

$4x_2+2x_3 = 0\\ 4x_1 +1x_2 = 0\\ 2x_1 +5x_3 = 0$

solving again we get

$x_1 = \dfrac{x_2}{4}= \dfrac{x_3}{8}$

$X_2 = k_2(1,4,8)$

Similarly, $X_3= k(1,8,4)$

$X_1.X_2 = X_1.X_3 = X_2.X_3= 0$

The normalised vector is

$A=\begin{vmatrix} -1/3& 2/3 &1/3\\ 4/3 & 1/3&8/3\\ 1/3&8/3&4/3 \end{vmatrix}$

b)

$a*b = sin(ab)$

Let a be 2 and b be 5

$sin(10°) = 0.174\\ sin(10)\ r = -0.544$

since, sinab $\neq$ N,

the operation is not binary