- Identify the dimensions of the first matrix \(A\) as \(2 \times 3\).
- Identify the dimensions of the second matrix \(B\) as \(3 \times 2\).
- Note that matrix multiplication \(AB\) is possible because the number of columns in \(A\) matches the number of rows in \(B\).
- The resulting matrix dimension is the number of rows of \(A\) by the number of columns of \(B\), which is \(\boxed{2 \times 2}\).
Explanation
- Analyze the dimensions of the input matrices
The problem asks for the dimensions of the product of two matrices, \(A\) and \(B\).
Matrix \(A\) is given as:
\[A = \begin{bmatrix} 2 & 4 & -3 \ 6 & 1 & 0 \end{bmatrix}\]
Matrix \(B\) is given as:
\[B = \begin{bmatrix} -3 & 2 \ -2 & 3 \ -8 & 5 \end{bmatrix}\]
To determine the dimensions of the resulting product matrix \(C = AB\), we look at the dimensions of \(A\) and \(B\). Matrix \(A\) has 2 rows and 3 columns, so its dimensions are \(2 \times 3\). Matrix \(B\) has 3 rows and 2 columns, so its dimensions are \(3 \times 2\).
- Determine the dimensions of the product matrix
Matrix multiplication \(C = AB\) is defined if and only if the number of columns in the first matrix (\(A\)) equals the number of rows in the second matrix (\(B\)).
Here, \(A\) has 3 columns and \(B\) has 3 rows. Since 3 = 3, the multiplication is valid.
The dimensions of the product matrix \(C\) are determined by the number of rows of the first matrix (\(A\)) and the number of columns of the second matrix (\(B\)).
Matrix \(A\) has 2 rows and Matrix \(B\) has 2 columns. Therefore, the resulting matrix \(C\) will have dimensions \(2 \times 2\).
- Conclusion
Given the calculation in the previous steps, we have confirmed that the dimensions of the product of a \(2 \times 3\) matrix and a \(3 \times 2\) matrix are \(2 \times 2\). Comparing this result with the provided options, the correct dimension is \(2 \times 2\).
Examples
Matrix multiplication is fundamental in computer graphics, where matrices represent transformations like rotation, scaling, and projection. For example, if you have a point in a 3D space represented as a \(1 \times 3\) vector, and you want to apply a sequence of transformations represented by a \(3 \times 3\) matrix, multiplying these ensures the correct transformation of the point. Understanding dimensions helps ensure that these operations are mathematically valid before performing complex calculations.