Sure! To find the product of the given matrices, follow these steps:
Consider the two matrices:
[tex]\[ A = \begin{pmatrix} -2 \\ 4 \end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix} 4 & 8 \end{pmatrix} \][/tex]
We need to perform matrix multiplication [tex]\( A \cdot B \)[/tex].
1. Set Up the Multiplication:
The matrix [tex]\( A \)[/tex] is a [tex]\( 2 \times 1 \)[/tex] matrix:
[tex]\[ A = \begin{pmatrix} -2 \\ 4 \end{pmatrix} \][/tex]
The matrix [tex]\( B \)[/tex] is a [tex]\( 1 \times 2 \)[/tex] matrix:
[tex]\[ B = \begin{pmatrix} 4 & 8 \end{pmatrix} \][/tex]
When multiplying two matrices, the resulting matrix [tex]\( C \)[/tex] will have dimensions corresponding to the number of rows of the first matrix and the number of columns of the second matrix. So, [tex]\( C \)[/tex] will be a [tex]\( 2 \times 2 \)[/tex] matrix.
2. Multiply and Sum the Elements:
To find the element [tex]\( c_{11} \)[/tex] in the first row and first column of matrix [tex]\( C \)[/tex], we multiply the corresponding elements from the first row of [tex]\( A \)[/tex] and the first column of [tex]\( B \)[/tex]:
[tex]\[ c_{11} = -2 \times 4 = -8 \][/tex]
To find the element [tex]\( c_{12} \)[/tex] in the first row and second column of matrix [tex]\( C \)[/tex], we multiply the corresponding elements from the first row of [tex]\( A \)[/tex] and the second column of [tex]\( B \)[/tex]:
[tex]\[ c_{12} = -2 \times 8 = -16 \][/tex]
To find the element [tex]\( c_{21} \)[/tex] in the second row and first column of matrix [tex]\( C \)[/tex], we multiply the corresponding elements from the second row of [tex]\( A \)[/tex] and the first column of [tex]\( B \)[/tex]:
[tex]\[ c_{21} = 4 \times 4 = 16 \][/tex]
To find the element [tex]\( c_{22} \)[/tex] in the second row and second column of matrix [tex]\( C \)[/tex], we multiply the corresponding elements from the second row of [tex]\( A \)[/tex] and the second column of [tex]\( B \)[/tex]:
[tex]\[ c_{22} = 4 \times 8 = 32 \][/tex]
3. Form the Resulting Matrix:
Combining these results, we get the resultant matrix [tex]\( C \)[/tex]:
[tex]\[ C = \begin{pmatrix} -8 & -16 \\ 16 & 32 \end{pmatrix} \][/tex]
So, the product of the given matrices is:
[tex]\[
\begin{pmatrix}
-8 & -16 \\
16 & 32
\end{pmatrix}
\][/tex]
