\[ \textbf{Q#2: Compute the Kronecker Product of the following matrices:} \] \[ A = \begin{pmatrix} 2 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix} \] \[ \text{Find } A \otimes B. \] \[ \textbf{Step 1: Formula for Kronecker Product} \] The Kronecker Product of \( A \otimes B \) is defined as: \[ A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \end{pmatrix} \] Here, \( A \) has \( 1 \times 2 \) dimensions, and \( B \) has \( 2 \times 2 \) dimensions. The resulting matrix will have dimensions: \[ (1 \times 2) \otimes (2 \times 2) = 2 \times 4 \] \[ \textbf{Step 2: Compute \( a_{ij} \cdot B \)} \] \[ \textbf{Compute } a_{11}B = 2 \cdot B: \] \[ a_{11}B = 2 \cdot \begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix} = \begin{pmatrix} 8 & 10 \\ 12 & 14 \end{pmatrix} \] \[ \textbf{Compute } a_{12}B = 3 \cdot B: \] \[ a_{12}B = 3 \cdot \begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix} = \begin{pmatrix} 12 & 15 \\ 18 & 21 \end{pmatrix} \] \[ \textbf{Step 3: Assemble the Resulting Matrix} \] Now combine the computed blocks \( a_{11}B \) and \( a_{12}B \) to form \( A \otimes B \): \[ A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \end{pmatrix} \] Substitute the blocks: \[ A \otimes B = \begin{pmatrix} 8 & 10 & 12 & 15 \\ 12 & 14 & 18 & 21 \end{pmatrix} \] \[ \textbf{Final Answer:} \] \[ A \otimes B = \begin{pmatrix} 8 & 10 & 12 & 15 \\ 12 & 14 & 18 & 21 \end{pmatrix} \]
Kronecker Product Exercise
\[\]
\[
\textbf{Q#1: Compute the Kronecker Product of the following matrices:}
\]
\[
A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 5 \\ 6 & 7 \end{pmatrix}
\]
\[
\textbf{Step 1: Use the formula for the Kronecker product.}
\]
\[
A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \end{pmatrix}
\]
\[
\textbf{Where each element of } A \textbf{ is multiplied by the entire matrix } B.
\]
\[
\textbf{Step 2: Substitute the values of } A \textbf{ and } B.
\]
\[
A \otimes B = \begin{pmatrix} 1 \cdot \begin{pmatrix} 0 & 5 \\ 6 & 7 \end{pmatrix} & 2 \cdot \begin{pmatrix} 0 & 5 \\ 6 & 7 \end{pmatrix} \\
3 \cdot \begin{pmatrix} 0 & 5 \\ 6 & 7 \end{pmatrix} & 4 \cdot \begin{pmatrix} 0 & 5 \\ 6 & 7 \end{pmatrix} \end{pmatrix}
\]
\[
\textbf{Step 3: Multiply each element.}
\]
\[
A \otimes B = \begin{pmatrix}
\begin{pmatrix} 0 & 5 \\ 6 & 7 \end{pmatrix} & \begin{pmatrix} 0 & 10 \\ 12 & 14 \end{pmatrix} \\
\begin{pmatrix} 0 & 15 \\ 18 & 21 \end{pmatrix} & \begin{pmatrix} 0 & 20 \\ 24 & 28 \end{pmatrix}
\end{pmatrix}
\]
\[
\textbf{Step 4: Combine into a single matrix.}
\]
\[
A \otimes B =
\begin{pmatrix}
0 & 5 & 0 & 10 \\
6 & 7 & 12 & 14 \\
0 & 15 & 0 & 20 \\
18 & 21 & 24 & 28
\end{pmatrix}
\]
\[
\textbf{Solution for Q1:}
\]
\[
A \otimes B =
\begin{pmatrix}
0 & 5 & 0 & 10 \\
6 & 7 & 12 & 14 \\
0 & 15 & 0 & 20 \\
18 & 21 & 24 & 28
\end{pmatrix}
\]
\[ \textbf{Q#2: Compute the Kronecker Product of the following matrices:} \] \[ A = \begin{pmatrix} 2 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix} \] \[ \text{Find } A \otimes B. \] \[ \textbf{Step 1: Formula for Kronecker Product} \] The Kronecker Product of \( A \otimes B \) is defined as: \[ A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \end{pmatrix} \] Here, \( A \) has \( 1 \times 2 \) dimensions, and \( B \) has \( 2 \times 2 \) dimensions. The resulting matrix will have dimensions: \[ (1 \times 2) \otimes (2 \times 2) = 2 \times 4 \] \[ \textbf{Step 2: Compute \( a_{ij} \cdot B \)} \] \[ \textbf{Compute } a_{11}B = 2 \cdot B: \] \[ a_{11}B = 2 \cdot \begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix} = \begin{pmatrix} 8 & 10 \\ 12 & 14 \end{pmatrix} \] \[ \textbf{Compute } a_{12}B = 3 \cdot B: \] \[ a_{12}B = 3 \cdot \begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix} = \begin{pmatrix} 12 & 15 \\ 18 & 21 \end{pmatrix} \] \[ \textbf{Step 3: Assemble the Resulting Matrix} \] Now combine the computed blocks \( a_{11}B \) and \( a_{12}B \) to form \( A \otimes B \): \[ A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \end{pmatrix} \] Substitute the blocks: \[ A \otimes B = \begin{pmatrix} 8 & 10 & 12 & 15 \\ 12 & 14 & 18 & 21 \end{pmatrix} \] \[ \textbf{Final Answer:} \] \[ A \otimes B = \begin{pmatrix} 8 & 10 & 12 & 15 \\ 12 & 14 & 18 & 21 \end{pmatrix} \]
\[ \textbf{Q#2: Compute the Kronecker Product of the following matrices:} \] \[ A = \begin{pmatrix} 2 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix} \] \[ \text{Find } A \otimes B. \] \[ \textbf{Step 1: Formula for Kronecker Product} \] The Kronecker Product of \( A \otimes B \) is defined as: \[ A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \end{pmatrix} \] Here, \( A \) has \( 1 \times 2 \) dimensions, and \( B \) has \( 2 \times 2 \) dimensions. The resulting matrix will have dimensions: \[ (1 \times 2) \otimes (2 \times 2) = 2 \times 4 \] \[ \textbf{Step 2: Compute \( a_{ij} \cdot B \)} \] \[ \textbf{Compute } a_{11}B = 2 \cdot B: \] \[ a_{11}B = 2 \cdot \begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix} = \begin{pmatrix} 8 & 10 \\ 12 & 14 \end{pmatrix} \] \[ \textbf{Compute } a_{12}B = 3 \cdot B: \] \[ a_{12}B = 3 \cdot \begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix} = \begin{pmatrix} 12 & 15 \\ 18 & 21 \end{pmatrix} \] \[ \textbf{Step 3: Assemble the Resulting Matrix} \] Now combine the computed blocks \( a_{11}B \) and \( a_{12}B \) to form \( A \otimes B \): \[ A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \end{pmatrix} \] Substitute the blocks: \[ A \otimes B = \begin{pmatrix} 8 & 10 & 12 & 15 \\ 12 & 14 & 18 & 21 \end{pmatrix} \] \[ \textbf{Final Answer:} \] \[ A \otimes B = \begin{pmatrix} 8 & 10 & 12 & 15 \\ 12 & 14 & 18 & 21 \end{pmatrix} \]