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To determine which matrices can be multiplied to the left of a vector matrix (a matrix with dimensions [tex]\( (2, 1) \)[/tex]), we need to understand the rules of matrix multiplication.
For matrix multiplication [tex]\( A \times B \)[/tex] to be valid, the number of columns in matrix [tex]\( A \)[/tex] must be equal to the number of rows in matrix [tex]\( B \)[/tex].
Given matrices:
A. [tex]\(\left[\begin{array}{l}6 \\ 7\end{array}\right]\)[/tex]
- Dimension: [tex]\( (2, 1) \)[/tex]
B. [tex]\(\left[\begin{array}{cc}-4 & 2 \\ 2 & 5 \\ 0 & 0\end{array}\right]\)[/tex]
- Dimension: [tex]\( (3, 2) \)[/tex]
C. [tex]\(\left[\begin{array}{cc}-1 & 2 \\ 5 & 3\end{array}\right]\)[/tex]
- Dimension: [tex]\( (2, 2) \)[/tex]
D. [tex]\(\left[\begin{array}{ll}-6 & 4\end{array}\right]\)[/tex]
- Dimension: [tex]\( (1, 2) \)[/tex]
We will examine each option to see if it can be multiplied to the left of a vector matrix with dimensions [tex]\( (2, 1) \)[/tex].
1. Option A:
- Dimension of matrix: [tex]\( (2, 1) \)[/tex]
- To multiply [tex]\((2, 1) \times (2, 1)\)[/tex]: The number of columns in the left matrix (2) does not match the number of rows in the right matrix (1).
- Thus, multiplication is not possible.
2. Option B:
- Dimension of matrix: [tex]\( (3, 2) \)[/tex]
- To multiply [tex]\((3, 2) \times (2, 1)\)[/tex]: The number of columns in the left matrix (2) matches the number of rows in the right matrix (2).
- Thus, multiplication is possible. The resulting matrix will have dimensions [tex]\( (3, 1) \)[/tex].
3. Option C:
- Dimension of matrix: [tex]\( (2, 2) \)[/tex]
- To multiply [tex]\((2, 2) \times (2, 1)\)[/tex]: The number of columns in the left matrix (2) matches the number of rows in the right matrix (2).
- Thus, multiplication is possible. The resulting matrix will have dimensions [tex]\( (2, 1) \)[/tex].
4. Option D:
- Dimension of matrix: [tex]\( (1, 2) \)[/tex]
- To multiply [tex]\((1, 2) \times (2, 1)\)[/tex]: The number of columns in the left matrix (2) matches the number of rows in the right matrix (2).
- Although the column-row number matches, multiplication produces a [tex]\( (1, 1) \)[/tex] result which does not result in a vector matrix with dimensions [tex]\( (2, 1) \)[/tex].
Therefore, the matrices that can be multiplied to the left of a vector matrix to produce a new vector matrix are:
Option B and Option C.
For matrix multiplication [tex]\( A \times B \)[/tex] to be valid, the number of columns in matrix [tex]\( A \)[/tex] must be equal to the number of rows in matrix [tex]\( B \)[/tex].
Given matrices:
A. [tex]\(\left[\begin{array}{l}6 \\ 7\end{array}\right]\)[/tex]
- Dimension: [tex]\( (2, 1) \)[/tex]
B. [tex]\(\left[\begin{array}{cc}-4 & 2 \\ 2 & 5 \\ 0 & 0\end{array}\right]\)[/tex]
- Dimension: [tex]\( (3, 2) \)[/tex]
C. [tex]\(\left[\begin{array}{cc}-1 & 2 \\ 5 & 3\end{array}\right]\)[/tex]
- Dimension: [tex]\( (2, 2) \)[/tex]
D. [tex]\(\left[\begin{array}{ll}-6 & 4\end{array}\right]\)[/tex]
- Dimension: [tex]\( (1, 2) \)[/tex]
We will examine each option to see if it can be multiplied to the left of a vector matrix with dimensions [tex]\( (2, 1) \)[/tex].
1. Option A:
- Dimension of matrix: [tex]\( (2, 1) \)[/tex]
- To multiply [tex]\((2, 1) \times (2, 1)\)[/tex]: The number of columns in the left matrix (2) does not match the number of rows in the right matrix (1).
- Thus, multiplication is not possible.
2. Option B:
- Dimension of matrix: [tex]\( (3, 2) \)[/tex]
- To multiply [tex]\((3, 2) \times (2, 1)\)[/tex]: The number of columns in the left matrix (2) matches the number of rows in the right matrix (2).
- Thus, multiplication is possible. The resulting matrix will have dimensions [tex]\( (3, 1) \)[/tex].
3. Option C:
- Dimension of matrix: [tex]\( (2, 2) \)[/tex]
- To multiply [tex]\((2, 2) \times (2, 1)\)[/tex]: The number of columns in the left matrix (2) matches the number of rows in the right matrix (2).
- Thus, multiplication is possible. The resulting matrix will have dimensions [tex]\( (2, 1) \)[/tex].
4. Option D:
- Dimension of matrix: [tex]\( (1, 2) \)[/tex]
- To multiply [tex]\((1, 2) \times (2, 1)\)[/tex]: The number of columns in the left matrix (2) matches the number of rows in the right matrix (2).
- Although the column-row number matches, multiplication produces a [tex]\( (1, 1) \)[/tex] result which does not result in a vector matrix with dimensions [tex]\( (2, 1) \)[/tex].
Therefore, the matrices that can be multiplied to the left of a vector matrix to produce a new vector matrix are:
Option B and Option C.
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