To determine whether the product [tex]\( AC \)[/tex] is defined, we need to check the dimensions of matrices [tex]\( A \)[/tex] and [tex]\( C \)[/tex].
Matrix [tex]\( A \)[/tex] is given by:
[tex]\[
A = \begin{pmatrix}
0 & 1 & -2 \\
1 & 6 & 4
\end{pmatrix}
\][/tex]
Matrix [tex]\( A \)[/tex] has dimensions [tex]\( 2 \times 3 \)[/tex] (2 rows and 3 columns).
Matrix [tex]\( C \)[/tex] is given by:
[tex]\[
C = \begin{pmatrix}
3 & 1 \\
5 & 5 \\
-2 & 5
\end{pmatrix}
\][/tex]
Matrix [tex]\( C \)[/tex] has dimensions [tex]\( 3 \times 2 \)[/tex] (3 rows and 2 columns).
In order for the matrix product [tex]\( AC \)[/tex] to be defined, the number of columns in matrix [tex]\( A \)[/tex] must be equal to the number of rows in matrix [tex]\( C \)[/tex]. Here, matrix [tex]\( A \)[/tex] has 3 columns and matrix [tex]\( C \)[/tex] has 3 rows, so the matrix product [tex]\( AC \)[/tex] is defined.
Next, we multiply matrices [tex]\( A \)[/tex] and [tex]\( C \)[/tex]. The resulting matrix will have dimensions [tex]\( 2 \times 2 \)[/tex] (the number of rows of [tex]\( A \)[/tex] and the number of columns of [tex]\( C \)[/tex]). The computation involves taking the dot product of rows of [tex]\( A \)[/tex] with the columns of [tex]\( C \)[/tex].
The resulting product [tex]\( AC \)[/tex] is:
[tex]\[
AC = \begin{pmatrix}
9 & -5 \\
25 & 51
\end{pmatrix}
\][/tex]
Thus, the correct choice is:
A. The expression [tex]\( AC \)[/tex] is defined. [tex]\( AC = \begin{pmatrix}
9 & -5 \\
25 & 51
\end{pmatrix} \)[/tex].
