Let's perform the matrix multiplication step-by-step to find the resulting vector matrix.
We are given two matrices:
[tex]\[ A = \begin{bmatrix} 6 & -5 \\ -3 & 4 \end{bmatrix} \][/tex]
and
[tex]\[ B = \begin{bmatrix} -1 \\ 3 \end{bmatrix} \][/tex]
We need to compute the product [tex]\( A \times B \)[/tex].
The product of a 2x2 matrix [tex]\( A \)[/tex] and a 2x1 matrix [tex]\( B \)[/tex] is a 2x1 matrix. The elements of this product matrix are computed as follows:
[tex]\[ \begin{bmatrix} 6 & -5 \\ -3 & 4 \end{bmatrix} \times \begin{bmatrix} -1 \\ 3 \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix} \][/tex]
Where:
[tex]\[ a = (6 \times -1) + (-5 \times 3) \][/tex]
[tex]\[ b = (-3 \times -1) + (4 \times 3) \][/tex]
Calculating each element:
[tex]\[ a = 6 \times -1 + (-5) \times 3 \][/tex]
[tex]\[ a = -6 + (-15) \][/tex]
[tex]\[ a = -6 - 15 \][/tex]
[tex]\[ a = -21 \][/tex]
Then,
[tex]\[ b = -3 \times -1 + 4 \times 3 \][/tex]
[tex]\[ b = 3 + 12 \][/tex]
[tex]\[ b = 15 \][/tex]
Thus, the resulting vector matrix is:
[tex]\[ \begin{bmatrix} -21 \\ 15 \end{bmatrix} \][/tex]
So, the values are:
[tex]\[ a = -21 \][/tex]
[tex]\[ b = 15 \][/tex]
