- Identify matrix \(A\) as a \(2 \times 2\) matrix.
- Identify matrix \(B\) as a \(2 \times 2\) matrix.
- Apply the matrix multiplication rule where an \(m \times n\) matrix multiplied by an \(n \times p\) matrix results in an \(m \times p\) matrix.
- Conclude that the product matrix \(C = AB\) is a \(2 \times 2\) matrix.
- Final answer: \(\boxed{2 \times 2}\)
Explanation
- Identify Dimensions of Matrices A and B
To determine the size of the product matrix \(C = AB\), we first identify the dimensions of the given matrices \(A\) and \(B\). Matrix \(A\) is given as \(\begin{bmatrix} 1 & -1 \ 0 & 3 \end{bmatrix}\), which consists of 2 rows and 2 columns, so it is a \(2 \times 2\) matrix. Matrix \(B\) is given as \(\begin{bmatrix} 0 & 2 \ 1 & -1 \end{bmatrix}\), which also consists of 2 rows and 2 columns, so it is a \(2 \times 2\) matrix.
- Check Matrix Multiplication Compatibility and Resulting Dimensions
The product of two matrices \(A\) and \(B\) is defined if the number of columns in \(A\) is equal to the number of rows in \(B\). If \(A\) is an \(m \times n\) matrix and \(B\) is an \(n \times p\) matrix, the resulting product matrix \(C = AB\) will have dimensions \(m \times p\). In our case, \(A\) is a \(2 \times 2\) matrix (\(m=2, n=2\)) and \(B\) is a \(2 \times 2\) matrix (\(n=2, p=2\)). Since the number of columns of \(A\) (which is 2) equals the number of rows of \(B\) (which is 2), the product \(C = AB\) is defined.
- Determine the Size of Product Matrix C
Based on the rule for matrix multiplication, since \(A\) is a \(2 \times 2\) matrix and \(B\) is a \(2 \times 2\) matrix, the resulting product matrix \(C\) will have the number of rows from \(A\) (2) and the number of columns from \(B\) (2). Therefore, the dimensions of the product matrix \(C\) are \(2 \times 2\).
Examples
In computer graphics, transformations of objects are represented by matrices. When you combine two transformations, such as rotating an object and then scaling it, you represent this by multiplying the matrices that correspond to each transformation. Knowing the dimensions of these matrices is essential to ensure that the mathematical operations are valid and to understand the dimensions of the final transformed space. For example, if you have a 2D coordinate vector represented as a \(2 \times 1\) matrix, multiplying it by a \(2 \times 2\) transformation matrix results in a new \(2 \times 1\) position. Understanding these dimensions helps in efficiently managing computations in real-time rendering and animation.