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Linear Algebra

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Difficult

Dot Product: Matrix Adjoint

LINALG-PVVGX3

Let $A=\begin{bmatrix} 1 & 2 & -1 \\\ 2 & 5 & -1 \\\ 1 & 2 & 0 \end{bmatrix}$

For all $\boldsymbol{u}, \ \boldsymbol{v} \in \mathbb{R}^3$, which matrix $B$ has the property that $(A\boldsymbol{u})\cdot \boldsymbol{v} = \boldsymbol{u}\cdot (B\boldsymbol{v})$?

A

$\begin{bmatrix} 2 & -2 & 3 \\\ -1 & 1 & -1 \\\ -1 & 0 & 1 \end{bmatrix}$

B

$\begin{bmatrix} 1 & 2 & 1 \\\ 2 & 5 & 2 \\\ -1 & -1 & 0 \end{bmatrix}$

C

$\begin{bmatrix} -1 & -2 & 1 \\\ -2 & -5 & 1 \\\ -1 & -2 & 0 \end{bmatrix}$

D

$\begin{bmatrix} -2 & 2 & -3 \\\ 1 & -1 & 1 \\\ 1 & 0 & -1 \end{bmatrix}$

E

$\begin{bmatrix} 2 & -1 & -1 \\\ -2 & 1 & 0 \\\ 3 & -1 & 1 \end{bmatrix}$

F

$\begin{bmatrix} -2 & 1 & 1 \\\ 2 & -1 & 0 \\\ -3 & 1 & -1 \end{bmatrix}$