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To determine which matrix multiplication is possible, we need to check the dimensions of the matrices involved in each multiplication. Specifically, for matrix multiplication [tex]\(A (m \times n)\)[/tex] and [tex]\(B (p \times q)\)[/tex] to be possible, the number of columns in [tex]\(A\)[/tex] (which is [tex]\(n\)[/tex]) must equal the number of rows in [tex]\(B\)[/tex] (which is [tex]\(p\)[/tex]).
Let's examine each pair of matrices:
1. [tex]\(\left[\begin{array}{ll}0 & 3\end{array}\right] \times\left[\begin{array}{ll}1 & -4\end{array}\right]\)[/tex]
- The first matrix [tex]\(\left[\begin{array}{ll}0 & 3\end{array}\right]\)[/tex] is a [tex]\(1 \times 2\)[/tex] matrix (1 row, 2 columns).
- The second matrix [tex]\(\left[\begin{array}{ll}1 & -4\end{array}\right]\)[/tex] is a [tex]\(1 \times 2\)[/tex] matrix (1 row, 2 columns).
- For multiplication to be possible, the number of columns in the first matrix (2) must equal the number of rows in the second matrix (1). Here, [tex]\(2 \neq 1\)[/tex], so this multiplication is not possible.
2. [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 [tex]\(\left[\begin{array}{c}3 \\ -2\end{array}\right]\)[/tex] is a [tex]\(2 \times 1\)[/tex] matrix (2 rows, 1 column).
- The second matrix [tex]\(\left[\begin{array}{cc}-1 & 0 \\ 0 & 3\end{array}\right]\)[/tex] is a [tex]\(2 \times 2\)[/tex] matrix (2 rows, 2 columns).
- For multiplication to be possible, the number of columns in the first matrix (1) must equal the number of rows in the second matrix (2). Here, [tex]\(1 \neq 2\)[/tex], so this multiplication is not possible.
3. [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 [tex]\(\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\)[/tex] is a [tex]\(2 \times 2\)[/tex] matrix (2 rows, 2 columns).
- The second matrix [tex]\(\left[\begin{array}{ll}3 & 0\end{array}\right]\)[/tex] is a [tex]\(1 \times 2\)[/tex] matrix (1 row, 2 columns).
- For multiplication to be possible, the number of columns in the first matrix (2) must equal the number of rows in the second matrix (1). Here, [tex]\(2 \neq 1\)[/tex], so this multiplication is not possible.
4. [tex]\(\left[\begin{array}{c}1 \\ -1\end{array}\right] \times\left[\begin{array}{ll}0 & 4\end{array}\right]\)[/tex]
- The first matrix [tex]\(\left[\begin{array}{c}1 \\ -1\end{array}\right]\)[/tex] is a [tex]\(2 \times 1\)[/tex] matrix (2 rows, 1 column).
- The second matrix [tex]\(\left[\begin{array}{ll}0 & 4\end{array}\right]\)[/tex] is a [tex]\(1 \times 2\)[/tex] matrix (1 row, 2 columns).
- For multiplication to be possible, the number of columns in the first matrix (1) must equal the number of rows in the second matrix (1). Here, [tex]\(1 = 1\)[/tex], so this multiplication is possible.
Thus, the only matrix multiplication that is possible is [tex]\(\left[\begin{array}{c}1 \\ -1\end{array}\right] \times\left[\begin{array}{ll}0 & 4\end{array}\right]\)[/tex].
Let's examine each pair of matrices:
1. [tex]\(\left[\begin{array}{ll}0 & 3\end{array}\right] \times\left[\begin{array}{ll}1 & -4\end{array}\right]\)[/tex]
- The first matrix [tex]\(\left[\begin{array}{ll}0 & 3\end{array}\right]\)[/tex] is a [tex]\(1 \times 2\)[/tex] matrix (1 row, 2 columns).
- The second matrix [tex]\(\left[\begin{array}{ll}1 & -4\end{array}\right]\)[/tex] is a [tex]\(1 \times 2\)[/tex] matrix (1 row, 2 columns).
- For multiplication to be possible, the number of columns in the first matrix (2) must equal the number of rows in the second matrix (1). Here, [tex]\(2 \neq 1\)[/tex], so this multiplication is not possible.
2. [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 [tex]\(\left[\begin{array}{c}3 \\ -2\end{array}\right]\)[/tex] is a [tex]\(2 \times 1\)[/tex] matrix (2 rows, 1 column).
- The second matrix [tex]\(\left[\begin{array}{cc}-1 & 0 \\ 0 & 3\end{array}\right]\)[/tex] is a [tex]\(2 \times 2\)[/tex] matrix (2 rows, 2 columns).
- For multiplication to be possible, the number of columns in the first matrix (1) must equal the number of rows in the second matrix (2). Here, [tex]\(1 \neq 2\)[/tex], so this multiplication is not possible.
3. [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 [tex]\(\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\)[/tex] is a [tex]\(2 \times 2\)[/tex] matrix (2 rows, 2 columns).
- The second matrix [tex]\(\left[\begin{array}{ll}3 & 0\end{array}\right]\)[/tex] is a [tex]\(1 \times 2\)[/tex] matrix (1 row, 2 columns).
- For multiplication to be possible, the number of columns in the first matrix (2) must equal the number of rows in the second matrix (1). Here, [tex]\(2 \neq 1\)[/tex], so this multiplication is not possible.
4. [tex]\(\left[\begin{array}{c}1 \\ -1\end{array}\right] \times\left[\begin{array}{ll}0 & 4\end{array}\right]\)[/tex]
- The first matrix [tex]\(\left[\begin{array}{c}1 \\ -1\end{array}\right]\)[/tex] is a [tex]\(2 \times 1\)[/tex] matrix (2 rows, 1 column).
- The second matrix [tex]\(\left[\begin{array}{ll}0 & 4\end{array}\right]\)[/tex] is a [tex]\(1 \times 2\)[/tex] matrix (1 row, 2 columns).
- For multiplication to be possible, the number of columns in the first matrix (1) must equal the number of rows in the second matrix (1). Here, [tex]\(1 = 1\)[/tex], so this multiplication is possible.
Thus, the only matrix multiplication that is possible is [tex]\(\left[\begin{array}{c}1 \\ -1\end{array}\right] \times\left[\begin{array}{ll}0 & 4\end{array}\right]\)[/tex].
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