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Determine whether [tex]\( AC \)[/tex] is defined. If it is defined, express the result as a single matrix; if it is not, say "not defined."

[tex]\[
A = \left[\begin{array}{rrr}
0 & 1 & -2 \\
1 & 6 & 4
\end{array}\right], \quad
B = \left[\begin{array}{rrr}
6 & 1 & 0 \\
-3 & 4 & -2 \\
\end{array}\right], \quad
C = \left[\begin{array}{rr}
3 & 1 \\
5 & 5 \\
-2 & 5
\end{array}\right]
\][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The expression [tex]\( AC \)[/tex] is defined. [tex]\( AC = \square \)[/tex] .

B. The expression [tex]\( AC \)[/tex] is not defined.



Answer :

GinnyAnswer
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].

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