To find [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] from the given matrix and vector multiplication, we will multiply the matrix by the vector step-by-step.
The given matrix is:
[tex]\[
\begin{bmatrix}
3 & 1 \\
-3 & 1
\end{bmatrix}
\][/tex]
The given vector is:
[tex]\[
\begin{bmatrix}
-2 \\
1
\end{bmatrix}
\][/tex]
We need to find the product of the matrix and the vector, which results in a new vector. To do this, we perform the matrix multiplication as follows:
1. Calculate [tex]\( x_1 \)[/tex]:
[tex]\[
x_1 = 3 \cdot (-2) + 1 \cdot 1
\][/tex]
2. Calculate [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = -3 \cdot (-2) + 1 \cdot 1
\][/tex]
Let's solve these step-by-step.
For [tex]\( x_1 \)[/tex]:
[tex]\[
x_1 = 3 \cdot (-2) + 1 \cdot 1 = -6 + 1 = -5
\][/tex]
For [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = -3 \cdot (-2) + 1 \cdot 1 = 6 + 1 = 7
\][/tex]
Thus, the resulting vector is:
[tex]\[
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} =
\begin{bmatrix}
-5 \\
7
\end{bmatrix}
\][/tex]
Therefore, we have:
[tex]\[
\begin{array}{l}
x_1 = -5 \\
x_2 = 7
\end{array}
\][/tex]
