Question

Compute the quadratic form xTAx for A = [3 2 0, 2 2 1, 0 1 0].

A. x = [x1 x2 x3].
B. x = [-2 -1 5].
C. x = [1/2 1/2 1/2].

Answer 1

(a)
`x^TAx=3x_1^2+4x_1x_2+2x_2^2+2x_2x_3`

(b) 12

(c)
`2(3)/(4)`

Explanation:

`G$iven A=\left[\begin{array}{ccc}3&2&0\\2&2&1\\0&1&0\end{array}\right]`

We are to compute the quadratic form
`x^TAx` for A.

Part A

`x=\left[\begin{array}{ccc}x_1\\x_2\\x_3\end{array}\right]`

`x^TAx = \left[\begin{array}{ccc}x_1&x_2&x_3\end{array}\right]\left[\begin{array}{ccc}3&2&0\\2&2&1\\0&1&0\end{array}\right]\left[\begin{array}{ccc}x_1\\x_2\\x_3\end{array}\right]`

`= \left[\begin{array}{ccc}x_1&x_2&x_3\end{array}\right]\left[\begin{array}{ccc}3x_1+2x_2+0x_3\\2x_1+2x_2+1x_3\\0x_1+1x_2+0x_3\end{array}\right]`

`= x_1(3x_1+2x_2+0x_3)+x_2(2x_1+2x_2+1x_3)+x_3(0x_1+1x_2+0x_3)\\= x_1(3x_1+2x_2)+x_2(2x_1+2x_2+x_3)+x_3(x_2)`

`= 3x_1^2+2x_1x_2+2x_1x_2+2x_2^2+x_2x_3+x_2x_3\\\\x^TAx=3x_1^2+4x_1x_2+2x_2^2+2x_2x_3`

Part B

`x=\left[\begin{array}{ccc}-2\\-1\\5\end{array}\right]\\x^TAx=3x_1^2+4x_1x_2+2x_2^2+2x_2x_3\\=3(-2)^2+4(-2)(-1)+2(-1)^2+2(-1)(5)\\=3*4+4*2+2-10\\=12+8+2-10\\=12`

Part C

`x=\left[\begin{array}{ccc}(1)/(2)\\(1)/(2)\\(1)/(2)\end{array}\right]\\x^TAx=3x_1^2+4x_1x_2+2x_2^2+2x_2x_3\\=3((1)/(2))^2+4((1)/(2))((1)/(2))+2((1)/(2))^2+2((1)/(2))((1)/(2))\\\\=(3)/(4)+1+ (1)/(2)+(1)/(2)\\\\=2(3)/(4)`