To determine which matrix can be multiplied to the left of a vector matrix to get a new vector matrix, we need to understand the rules of matrix multiplication.
1. Matrix Dimensions and Multiplication:
- A vector matrix typically is a column vector, represented as an [tex]\( n \times 1 \)[/tex] matrix.
- Matrix multiplication is defined if the number of columns in the left matrix (first matrix) matches the number of rows in the right matrix (second matrix).
Given matrices:
- Matrix A: [tex]\( \left[\begin{array}{l}6 \\ 7\end{array}\right] \)[/tex] (2x1 matrix)
- Matrix B: [tex]\( \left[\begin{array}{cc}-4 & 2 \\ 2 & 5 \\ 0 & 0\end{array}\right] \)[/tex] (3x2 matrix)
- Matrix C: [tex]\( \left[\begin{array}{cc}-1 & 2 \\ 5 & 3\end{array}\right] \)[/tex] (2x2 matrix)
- Matrix D: [tex]\( \left[\begin{array}{ll}-6 & 4\end{array}\right] \)[/tex] (1x2 matrix)
2. Check Each Matrix for Multiplicative Compatibility with a Vector Matrix:
- A vector matrix, in this context, can be assumed to be a column vector, e.g., [tex]\( \left[\begin{array}{c} x \\ y \end{array}\right] \)[/tex] which is a [tex]\( 2 \times 1 \)[/tex] matrix.
Matrix A: [tex]\( 2 \times 1 \)[/tex]
- Matrix A has 2 rows and 1 column.
- Matrix A cannot multiply with another [tex]\( 2 \times 1 \)[/tex] vector matrix on the left because [tex]\( 2 \times 1 \)[/tex] cannot be multiplied with [tex]\( 2 \times 1 \)[/tex].
Matrix B: [tex]\( 3 \times 2 \)[/tex]
- Matrix B has 3 rows and 2 columns.
- Matrix B can multiply a [tex]\( 2 \times 1 \)[/tex] vector matrix (column vector) because the number of columns in B (2) matches the number of rows in the column vector (2).
- The resulting matrix will be a [tex]\( 3 \times 1 \)[/tex] vector matrix.
Matrix C: [tex]\( 2 \times 2 \)[/tex]
- Matrix C has 2 rows and 2 columns.
- Matrix C can multiply a [tex]\( 2 \times 1 \)[/tex] vector matrix because the number of columns in C (2) matches the number of rows in the column vector (2).
- The resulting matrix will be a [tex]\( 2 \times 1 \)[/tex] vector matrix.
Matrix D: [tex]\( 1 \times 2 \)[/tex]
- Matrix D has 1 row and 2 columns.
- Matrix D cannot multiply with a [tex]\( 2 \times 1 \)[/tex] vector matrix on the left because [tex]\( 1 \times 2 \)[/tex] cannot be multiplied with [tex]\( 2 \times 1 \)[/tex].
3. Conclusion:
- Matrix B ([tex]\( 3 \times 2 \)[/tex]) can be multiplied to the left of a [tex]\( 2 \times 1 \)[/tex] vector matrix to get a new vector matrix.
Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
