To solve the given question, follow these steps for finding the products of the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
### Determine the size of the product [tex]\( C = AB \)[/tex]
Matrix [tex]\( A \)[/tex] is a [tex]\( 2 \times 2 \)[/tex] matrix and matrix [tex]\( B \)[/tex] is also a [tex]\( 2 \times 2 \)[/tex] matrix. When you multiply two matrices, the resulting matrix has dimensions corresponding to the number of rows of the first matrix and the number of columns of the second matrix.
Since both [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are [tex]\( 2 \times 2 \)[/tex], the resulting matrix [tex]\( C \)[/tex] will also be [tex]\( 2 \times 2 \)[/tex].
### Calculate the product [tex]\( C = AB \)[/tex]
To find the elements of matrix [tex]\( C \)[/tex]:
- [tex]\( c_{11} \)[/tex] is the dot product of the first row of [tex]\( A \)[/tex] and the first column of [tex]\( B \)[/tex].
- [tex]\( c_{12} \)[/tex] is the dot product of the first row of [tex]\( A \)[/tex] and the second column of [tex]\( B \)[/tex].
- [tex]\( c_{21} \)[/tex] is the dot product of the second row of [tex]\( A \)[/tex] and the first column of [tex]\( B \)[/tex].
- [tex]\( c_{22} \)[/tex] is the dot product of the second row of [tex]\( A \)[/tex] and the second column of [tex]\( B \)[/tex].
Given the result:
[tex]\[ c_{11} = -1, \][/tex]
[tex]\[ c_{12} = 3. \][/tex]
### Calculate the product [tex]\( D = BA \)[/tex]
To find the elements of matrix [tex]\( D \)[/tex]:
- [tex]\( d_{11} \)[/tex] is the dot product of the first row of [tex]\( B \)[/tex] and the first column of [tex]\( A \)[/tex].
- [tex]\( d_{12} \)[/tex] is the dot product of the first row of [tex]\( B \)[/tex] and the second column of [tex]\( A \)[/tex].
- [tex]\( d_{21} \)[/tex] is the dot product of the second row of [tex]\( B \)[/tex] and the first column of [tex]\( A \)[/tex].
- [tex]\( d_{22} \)[/tex] is the dot product of the second row of [tex]\( B \)[/tex] and the second column of [tex]\( A \)[/tex].
Given the results:
[tex]\[ d_{11} = 0, \][/tex]
[tex]\[ d_{12} = 6, \][/tex]
[tex]\[ d_{21} = 1, \][/tex]
[tex]\[ d_{22} = -4. \][/tex]
### Final Answers:
The size of the product [tex]\( C = AB \)[/tex]:
[tex]\[
\begin{array}{c|c}
\checkmark & \times 2 \\
\hline
\square & \square \checkmark \\
\end{array}
\][/tex]
Product [tex]\( C = AB \)[/tex]:
[tex]\[
c_{11} = -1, \quad c_{12} = 3.
\][/tex]
Product [tex]\( D = BA \)[/tex]:
\[
\begin{array}{l}
d_{11} = 0 \\
d_{12} = 6 \\
d_{21} = 1 \\
d_{22} = -4 \\
\end{array}
]
]
These results correctly reflect the products of the given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
