To find the product of matrix [tex]\( \mathbf{A} \)[/tex] and vector [tex]\( \mathbf{B} \)[/tex], we need to perform matrix multiplication. Here is a step-by-step process:
Given:
[tex]\[
\mathbf{A} = \begin{pmatrix}
-2 & 3 & 6 \\
4 & -1 & 10
\end{pmatrix}
\][/tex]
[tex]\[
\mathbf{B} = \begin{pmatrix}
-3 \\
3 \\
9
\end{pmatrix}
\][/tex]
We want to calculate the product [tex]\( \mathbf{A} \cdot \mathbf{B} \)[/tex].
### Step-by-Step Solution:
#### Step 1: Understand the matrix dimensions
Matrix [tex]\( \mathbf{A} \)[/tex] is a [tex]\( 2 \times 3 \)[/tex] matrix, and vector [tex]\( \mathbf{B} \)[/tex] is a [tex]\( 3 \times 1 \)[/tex] matrix. Since the number of columns in [tex]\( \mathbf{A} \)[/tex] matches the number of rows in [tex]\( \mathbf{B} \)[/tex], matrix multiplication is defined.
#### Step 2: Perform multiplication
Each element of the resulting matrix (which will be a [tex]\( 2 \times 1 \)[/tex] matrix) is computed as follows:
[tex]\[
\mathbf{C} = \mathbf{A} \cdot \mathbf{B} = \begin{pmatrix}
c_{11} \\
c_{21}
\end{pmatrix}
\][/tex]
Where each element [tex]\( c_{ij} \)[/tex] is calculated by taking the dot product of the [tex]\( i \)[/tex]-th row of [tex]\( \mathbf{A} \)[/tex] with the [tex]\( j \)[/tex]-th column of [tex]\( \mathbf{B} \)[/tex].
For [tex]\( c_{11} \)[/tex]:
[tex]\[
c_{11} = (-2 \times -3) + (3 \times 3) + (6 \times 9)
\][/tex]
For [tex]\( c_{21} \)[/tex]:
[tex]\[
c_{21} = (4 \times -3) + (-1 \times 3) + (10 \times 9)
\][/tex]
#### Step 3: Simplify each element
Calculate [tex]\( c_{11} \)[/tex]:
[tex]\[
c_{11} = 6 + 9 + 54 = 69
\][/tex]
Calculate [tex]\( c_{21} \)[/tex]:
[tex]\[
c_{21} = -12 - 3 + 90 = 75
\][/tex]
#### Step 4: Form the resulting product matrix
Thus, the resulting product matrix is:
[tex]\[
\mathbf{C} = \begin{pmatrix}
69 \\
75
\end{pmatrix}
\][/tex]
So the product of matrix [tex]\( \mathbf{A} \)[/tex] and vector [tex]\( \mathbf{B} \)[/tex] is:
[tex]\[
\boxed{\begin{pmatrix}
69 \\
75
\end{pmatrix}}
\][/tex]
