To calculate the product of the matrix [tex]\( A \)[/tex] and vector [tex]\( B \)[/tex], we need to perform matrix multiplication. Specifically, we'll multiply each row of matrix [tex]\( A \)[/tex] by vector [tex]\( B \)[/tex].
The matrix [tex]\( A \)[/tex] is:
[tex]\[
A = \begin{pmatrix}
0 & -1 \\
5 & 4
\end{pmatrix}
\][/tex]
The vector [tex]\( B \)[/tex] is:
[tex]\[
B = \begin{pmatrix}
1 \\
1
\end{pmatrix}
\][/tex]
For matrix multiplication, the resulting vector [tex]\( C \)[/tex] will be:
[tex]\[
C = A \cdot B
\][/tex]
Now, let's calculate each element of the resulting vector [tex]\( C \)[/tex].
1. First element [tex]\( c_1 \)[/tex]:
[tex]\[
c_1 = 0 \cdot 1 + (-1) \cdot 1 = 0 - 1 = -1
\][/tex]
2. Second element [tex]\( c_2 \)[/tex]:
[tex]\[
c_2 = 5 \cdot 1 + 4 \cdot 1 = 5 + 4 = 9
\][/tex]
Putting these elements together, the resulting vector [tex]\( C \)[/tex] is:
[tex]\[
C = \begin{pmatrix}
-1 \\
9
\end{pmatrix}
\][/tex]
So, the product of the given matrix and vector is:
[tex]\[
\begin{pmatrix}
0 & -1 \\
5 & 4
\end{pmatrix}
\begin{pmatrix}
1 \\
1
\end{pmatrix}
=
\begin{pmatrix}
-1 \\
9
\end{pmatrix}
\][/tex]
Therefore, the answer is:
[tex]\[
\begin{pmatrix}
-1 \\
9
\end{pmatrix}
\][/tex]
