To determine the dimensions of the product of two matrices, we need to recall the rules of matrix multiplication. Specifically, the product of an [tex]\( m \times n \)[/tex] matrix and an [tex]\( n \times p \)[/tex] matrix results in an [tex]\( m \times p \)[/tex] matrix. That is, the number of rows in the resulting matrix will be the same as the number of rows in the first matrix, and the number of columns will be the same as the number of columns in the second matrix.
Let's examine the given matrices:
[tex]\[
A = \begin{bmatrix}
2 & 4 & -3 \\
6 & 1 & 0
\end{bmatrix}
\][/tex]
[tex]\[
B = \begin{bmatrix}
-3 & 2 \\
-2 & 3 \\
-8 & 5
\end{bmatrix}
\][/tex]
1. Determine the dimensions of [tex]\( A \)[/tex]:
- [tex]\( A \)[/tex] has 2 rows and 3 columns, so [tex]\( A \)[/tex] is a [tex]\( 2 \times 3 \)[/tex] matrix.
2. Determine the dimensions of [tex]\( B \)[/tex]:
- [tex]\( B \)[/tex] has 3 rows and 2 columns, so [tex]\( B \)[/tex] is a [tex]\( 3 \times 2 \)[/tex] matrix.
To multiply [tex]\( A \)[/tex] by [tex]\( B \)[/tex], the number of columns in [tex]\( A \)[/tex] must equal the number of rows in [tex]\( B \)[/tex]. In this case, both values are 3, so the multiplication is defined.
3. Determine the dimensions of the resulting matrix:
- The resulting matrix will have the same number of rows as [tex]\( A \)[/tex] (2 rows) and the same number of columns as [tex]\( B \)[/tex] (2 columns).
Therefore, the dimensions of the product matrix are [tex]\( 2 \times 2 \)[/tex].
To summarize, the correct answer is:
[tex]\[ 2 \times 2 \][/tex]
