Let’s carefully solve the matrix multiplication step-by-step for the given matrices:
Given matrices:
[tex]\[ A = \begin{bmatrix} -1 & 5 \\ 7 & 0 \end{bmatrix} \][/tex]
and
[tex]\[ B = \begin{bmatrix} 5 \\ 0 \end{bmatrix} \][/tex]
We need to multiply matrix [tex]\( A \)[/tex] by matrix [tex]\( B \)[/tex], which can be achieved by performing the dot product of rows of [tex]\( A \)[/tex] with the column of [tex]\( B \)[/tex].
The resulting matrix, let's call it matrix [tex]\( C \)[/tex], will be a 2x1 matrix since [tex]\( A \)[/tex] is a 2x2 matrix and [tex]\( B \)[/tex] is a 2x1 matrix.
So, matrix [tex]\( C \)[/tex] will be,
[tex]\[ C = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} \][/tex]
Where:
[tex]\[ c_1 = (-1) \cdot 5 + 5 \cdot 0 \][/tex]
[tex]\[ c_2 = 7 \cdot 5 + 0 \cdot 0 \][/tex]
Calculate [tex]\( c_1 \)[/tex]:
[tex]\[ c_1 = (-1) \cdot 5 + 5 \cdot 0 \][/tex]
[tex]\[ c_1 = -5 + 0 \][/tex]
[tex]\[ c_1 = -5 \][/tex]
Calculate [tex]\( c_2 \)[/tex]:
[tex]\[ c_2 = 7 \cdot 5 + 0 \cdot 0 \][/tex]
[tex]\[ c_2 = 35 + 0 \][/tex]
[tex]\[ c_2 = 35 \][/tex]
Thus, the resulting matrix [tex]\( C \)[/tex] is:
[tex]\[ C = \begin{bmatrix} -5 \\ 35 \end{bmatrix} \][/tex]
Therefore, the answer to the matrix multiplication [tex]\(\left[\begin{array}{cc}-1 & 5 \\ 7 & 0\end{array}\right]\left[\begin{array}{l}5 \\ 0\end{array}\right]\)[/tex] is:
[tex]\[ \begin{bmatrix} -5 \\ 35 \end{bmatrix} \][/tex]
