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To determine which matrix can be multiplied to the left of a vector matrix to obtain a new vector matrix, we need to consider matrix dimensions and the rules for matrix multiplication.
- A vector matrix typically has dimensions [tex]\( n \times 1 \)[/tex].
- For a matrix [tex]\( A \)[/tex] to be multiplied to the left of a vector matrix [tex]\( x \)[/tex], [tex]\( A \)[/tex] must have dimensions [tex]\( m \times n \)[/tex], where [tex]\( n \)[/tex] is the number of rows in the vector matrix [tex]\( x \)[/tex], and [tex]\( m \)[/tex] is the resulting vector matrix dimension.
Let's consider the vector matrix [tex]\( x \)[/tex] as having dimensions:
[tex]\[ x = \left[\begin{array}{c} x_1 \\ x_2 \end{array}\right] \quad \text{(a 2x1 vector)}\][/tex]
Now, let's analyze each given matrix:
A. [tex]\[ \left[\begin{array}{l} 6 \\ 7 \end{array}\right] \][/tex]
This is a [tex]\( 2 \times 1 \)[/tex] matrix. Multiplying this by a 2x1 vector is not possible as the dimensions do not align.
B. [tex]\[ \left[\begin{array}{cc} -4 & 2 \\ 2 & 5 \\ 0 & 0 \end{array}\right] \][/tex]
This is a [tex]\( 3 \times 2 \)[/tex] matrix. Multiplying this by a 2x1 vector is possible because the number of columns in the matrix (2) matches the number of rows in the vector matrix (2). The resulting matrix will be a [tex]\( 3 \times 1 \)[/tex] vector matrix.
C. [tex]\[ \left[\begin{array}{cc} -1 & 2 \\ 5 & 3 \end{array}\right] \][/tex]
This is a [tex]\( 2 \times 2 \)[/tex] matrix. Multiplying this by a 2x1 vector is possible because the number of columns in the matrix (2) matches the number of rows in the vector matrix (2). The resulting matrix will be a [tex]\( 2 \times 1 \)[/tex] vector matrix.
D. [tex]\[ \left[\begin{array}{ll} -6 & 4 \end{array}\right] \][/tex]
This is a [tex]\( 1 \times 2 \)[/tex] matrix. Multiplying this by a 2x1 vector is possible because the number of columns in the matrix (2) matches the number of rows in the vector matrix (2). The resulting matrix will be a [tex]\( 1 \times 1 \)[/tex] matrix, which is not a vector matrix.
Hence, the matrices that can be multiplied to the left of a vector matrix to get a new vector matrix are:
- Option B (as a [tex]\( 3 \times 1 \)[/tex] vector matrix result)
- Option C (as a [tex]\( 2 \times 1 \)[/tex] vector matrix result)
Thus, the correct answers are:
B. [tex]\(\left[\begin{array}{cc} -4 & 2 \\ 2 & 5 \\ 0 & 0 \end{array}\right]\)[/tex]
C. [tex]\(\left[\begin{array}{cc} -1 & 2 \\ 5 & 3 \end{array}\right]\)[/tex]
- A vector matrix typically has dimensions [tex]\( n \times 1 \)[/tex].
- For a matrix [tex]\( A \)[/tex] to be multiplied to the left of a vector matrix [tex]\( x \)[/tex], [tex]\( A \)[/tex] must have dimensions [tex]\( m \times n \)[/tex], where [tex]\( n \)[/tex] is the number of rows in the vector matrix [tex]\( x \)[/tex], and [tex]\( m \)[/tex] is the resulting vector matrix dimension.
Let's consider the vector matrix [tex]\( x \)[/tex] as having dimensions:
[tex]\[ x = \left[\begin{array}{c} x_1 \\ x_2 \end{array}\right] \quad \text{(a 2x1 vector)}\][/tex]
Now, let's analyze each given matrix:
A. [tex]\[ \left[\begin{array}{l} 6 \\ 7 \end{array}\right] \][/tex]
This is a [tex]\( 2 \times 1 \)[/tex] matrix. Multiplying this by a 2x1 vector is not possible as the dimensions do not align.
B. [tex]\[ \left[\begin{array}{cc} -4 & 2 \\ 2 & 5 \\ 0 & 0 \end{array}\right] \][/tex]
This is a [tex]\( 3 \times 2 \)[/tex] matrix. Multiplying this by a 2x1 vector is possible because the number of columns in the matrix (2) matches the number of rows in the vector matrix (2). The resulting matrix will be a [tex]\( 3 \times 1 \)[/tex] vector matrix.
C. [tex]\[ \left[\begin{array}{cc} -1 & 2 \\ 5 & 3 \end{array}\right] \][/tex]
This is a [tex]\( 2 \times 2 \)[/tex] matrix. Multiplying this by a 2x1 vector is possible because the number of columns in the matrix (2) matches the number of rows in the vector matrix (2). The resulting matrix will be a [tex]\( 2 \times 1 \)[/tex] vector matrix.
D. [tex]\[ \left[\begin{array}{ll} -6 & 4 \end{array}\right] \][/tex]
This is a [tex]\( 1 \times 2 \)[/tex] matrix. Multiplying this by a 2x1 vector is possible because the number of columns in the matrix (2) matches the number of rows in the vector matrix (2). The resulting matrix will be a [tex]\( 1 \times 1 \)[/tex] matrix, which is not a vector matrix.
Hence, the matrices that can be multiplied to the left of a vector matrix to get a new vector matrix are:
- Option B (as a [tex]\( 3 \times 1 \)[/tex] vector matrix result)
- Option C (as a [tex]\( 2 \times 1 \)[/tex] vector matrix result)
Thus, the correct answers are:
B. [tex]\(\left[\begin{array}{cc} -4 & 2 \\ 2 & 5 \\ 0 & 0 \end{array}\right]\)[/tex]
C. [tex]\(\left[\begin{array}{cc} -1 & 2 \\ 5 & 3 \end{array}\right]\)[/tex]
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