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To determine which matrix multiplications are possible, we need to check the compatibility of the dimensions of the matrices involved. Specifically, matrix multiplication is possible when the number of columns in the first matrix matches the number of rows in the second matrix.
1. First Matrix Multiplication:
[tex]\[ \left[\begin{array}{ll}0 & 3\end{array}\right] \times \left[\begin{array}{ll}1 & -4\end{array}\right] \][/tex]
The first matrix is [tex]\( 1 \times 2 \)[/tex] (1 row, 2 columns) and the second matrix is also [tex]\( 1 \times 2 \)[/tex] (1 row, 2 columns). Since the number of columns in the first matrix (2) does not match the number of rows in the second matrix (1), this matrix multiplication is not possible. So, the result for this multiplication is 0.
2. Second Matrix Multiplication:
[tex]\[ \left[\begin{array}{c}3 \\ -2\end{array}\right] \times \left[\begin{array}{cc}-1 & 0 \\ 0 & 3\end{array}\right] \][/tex]
The first matrix is [tex]\( 2 \times 1 \)[/tex] (2 rows, 1 column) and the second matrix is [tex]\( 2 \times 2 \)[/tex] (2 rows, 2 columns). The number of columns in the first matrix (1) does not match the number of rows in the second matrix (2), so this matrix multiplication is not possible. So, the result for this multiplication is 0.
3. Third Matrix Multiplication:
[tex]\[ \left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right] \times \left[\begin{array}{ll}3 & 0\end{array}\right] \][/tex]
The first matrix is [tex]\( 2 \times 2 \)[/tex] (2 rows, 2 columns) and the second matrix is [tex]\( 1 \times 2 \)[/tex] (1 row, 2 columns). The number of columns in the first matrix (2) does not match the number of rows in the second matrix (1), so this matrix multiplication is not possible. So, the result for this multiplication is 0.
4. Fourth Matrix Multiplication:
[tex]\[ \left[\begin{array}{c}1 \\ -1\end{array}\right] \times \left[\begin{array}{ll}0 & 4\end{array}\right] \][/tex]
The first matrix is [tex]\( 2 \times 1 \)[/tex] (2 rows, 1 column) and the second matrix is [tex]\( 1 \times 2 \)[/tex] (1 row, 2 columns). The number of columns in the first matrix (1) matches the number of rows in the second matrix (1), so this matrix multiplication is possible. Thus, the result for this multiplication is 1.
Combining all the results, the possible matrix multiplications are as follows:
[tex]\[ (0, 0, 0, 1) \][/tex]
This means that among the given matrix multiplications, only the last one is possible.
1. First Matrix Multiplication:
[tex]\[ \left[\begin{array}{ll}0 & 3\end{array}\right] \times \left[\begin{array}{ll}1 & -4\end{array}\right] \][/tex]
The first matrix is [tex]\( 1 \times 2 \)[/tex] (1 row, 2 columns) and the second matrix is also [tex]\( 1 \times 2 \)[/tex] (1 row, 2 columns). Since the number of columns in the first matrix (2) does not match the number of rows in the second matrix (1), this matrix multiplication is not possible. So, the result for this multiplication is 0.
2. Second Matrix Multiplication:
[tex]\[ \left[\begin{array}{c}3 \\ -2\end{array}\right] \times \left[\begin{array}{cc}-1 & 0 \\ 0 & 3\end{array}\right] \][/tex]
The first matrix is [tex]\( 2 \times 1 \)[/tex] (2 rows, 1 column) and the second matrix is [tex]\( 2 \times 2 \)[/tex] (2 rows, 2 columns). The number of columns in the first matrix (1) does not match the number of rows in the second matrix (2), so this matrix multiplication is not possible. So, the result for this multiplication is 0.
3. Third Matrix Multiplication:
[tex]\[ \left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right] \times \left[\begin{array}{ll}3 & 0\end{array}\right] \][/tex]
The first matrix is [tex]\( 2 \times 2 \)[/tex] (2 rows, 2 columns) and the second matrix is [tex]\( 1 \times 2 \)[/tex] (1 row, 2 columns). The number of columns in the first matrix (2) does not match the number of rows in the second matrix (1), so this matrix multiplication is not possible. So, the result for this multiplication is 0.
4. Fourth Matrix Multiplication:
[tex]\[ \left[\begin{array}{c}1 \\ -1\end{array}\right] \times \left[\begin{array}{ll}0 & 4\end{array}\right] \][/tex]
The first matrix is [tex]\( 2 \times 1 \)[/tex] (2 rows, 1 column) and the second matrix is [tex]\( 1 \times 2 \)[/tex] (1 row, 2 columns). The number of columns in the first matrix (1) matches the number of rows in the second matrix (1), so this matrix multiplication is possible. Thus, the result for this multiplication is 1.
Combining all the results, the possible matrix multiplications are as follows:
[tex]\[ (0, 0, 0, 1) \][/tex]
This means that among the given matrix multiplications, only the last one is possible.
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