$$\text { Compute the quadratic form } x ^ { T } A x \text { for the given matrix } A \text { and vector } x \text {. }$$
-$$A = \left[ \begin{array} { r r r }
4 & 8 & 0 \
8 & 1 & 2 \
0 & 2 & - 2
\end{array} \right] , x = \left[ \begin{array} { l }
x _ { 1 } \
x _ { 2 } \
x _ { 3 }
\end{array} \right]$$
A) $4 x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } - 2 x _ { 3 } ^ { 2 } + 8 x _ { 1 } + 10 x _ { 2 } + 2 x _ { 3 }$
B) $4 \times \frac { 2 } { 1 } + x _ { 2 } ^ { 2 } - 2 \times \frac { 2 } { 3 }$
C) $4 x _ { 1 } ^ { 2 } - x _ { 2 } ^ { 2 } + 2 x _ { 3 } ^ { 2 } - 16 x _ { 1 } x _ { 2 } - 4 x _ { 2 } x _ { 3 }$
D) $4 x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } - 2 x _ { 3 } ^ { 2 } + 16 x _ { 1 } x _ { 2 } + 4 x _ { 2 } x _ { 3 }$

Question

$$\text { Compute the quadratic form } x ^ { T } A x \text { for the given matrix } A \text { and vector } x \text {. }$$
-$$A = \left[ \begin{array} { r r r } 
4 & 8 & 0 \
8 & 1 & 2 \
0 & 2 & - 2
\end{array} \right] , x = \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 }
\end{array} \right]$$
A) $4 x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } - 2 x _ { 3 } ^ { 2 } + 8 x _ { 1 } + 10 x _ { 2 } + 2 x _ { 3 }$
B) $4 \times \frac { 2 } { 1 } + x _ { 2 } ^ { 2 } - 2 \times \frac { 2 } { 3 }$
C) $4 x _ { 1 } ^ { 2 } - x _ { 2 } ^ { 2 } + 2 x _ { 3 } ^ { 2 } - 16 x _ { 1 } x _ { 2 } - 4 x _ { 2 } x _ { 3 }$
D) $4 x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } - 2 x _ { 3 } ^ { 2 } + 16 x _ { 1 } x _ { 2 } + 4 x _ { 2 } x _ { 3 }$

Quizplus · Accepted Answer

The quadratic form $x^TAx$ for a given matrix $A$ and vector $x$ is calculated by substituting the elements of $A$ and $x$ into the formula. The correct calculation leads to $4x_1^2 + x_2^2 - 2x_3^2 + 16x_1x_2 + 4x_2x_3$, which matches option D. This includes the squared terms, the cross terms multiplied by their respective coefficients from the matrix $A$, and accounts for the symmetry in $A$ where the off-diagonal elements contribute to the cross terms.

Compute the quadratic form xTAx for the given matrix A and vector x. \text { Compute the quadratic form } x ^ { T } A x \text { for the given matrix } A \text { and vector } x \text {. } Compute the quadratic form xTAx for the given matrix A and vector x.

$\text { Compute the quadratic form } x ^ { T } A x \text { for the given matrix } A \text { and vector } x \text {. }$