Let \( \mathbf{A}=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right] \) and \( \mathbf{B}=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] \) Identify \( \mathbf{A} \odot \mathbf{B} \).
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To find the element-wise product \( \mathbf{A} \odot \mathbf{B} \), we multiply corresponding elements of matrices \( \mathbf{A} \) and \( \mathbf{B} \). Given: \[ \mathbf{A}=\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}, \quad \mathbf{B}=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} \] Now, we calculate \( \mathbf{A} \odot \mathbf{B} \): \[ \mathbf{A} \odot \mathbf{B} = \begin{bmatrix} 1 \cdot 0 & 1 \cdot 1 \\ 0 \cdot 1 & 1 \cdot 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \] Thus, the element-wise product \( \mathbf{A} \odot \mathbf{B} \) is: \[ \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \]