- Identify that the product of two \(2 \times 2\) matrices is another \(2 \times 2\) matrix.
- Calculate each entry using the dot product of rows and columns:
- \(c_{11} = (1 \times 0) + (-1 \times 1) = -1\)
- \(c_{12} = (1 \times 2) + (-1 \times -1) = 3\)
- \(c_{21} = (0 \times 0) + (3 \times 1) = 3\)
- \(c_{22} = (0 \times 2) + (3 \times -1) = -3\)
- The resulting matrix is \(C = \begin{bmatrix} -1 & 3 \ 3 & -3 \end{bmatrix}\).
- Final Answer: \(\boxed{C = \begin{bmatrix} -1 & 3 \ 3 & -3 \end{bmatrix}}\)
Explanation
- Analyzing Dimensions of the Matrices
To find the product of two matrices \(A\) and \(B\), we first identify their dimensions. Here, matrix \(A = \begin{bmatrix} 1 & -1 \ 0 & 3 \end{bmatrix}\) is a \(2 \times 2\) matrix, and matrix \(B = \begin{bmatrix} 0 & 2 \ 1 & -1 \end{bmatrix}\) is also a \(2 \times 2\) matrix. Since the number of columns in \(A\) (which is 2) matches the number of rows in \(B\) (which is 2), the product \(C = AB\) is well-defined. The resulting matrix \(C\) will have dimensions corresponding to the number of rows of \(A\) and the number of columns of \(B\), so \(C\) is a \(2 \times 2\) matrix.
- Calculating the Entries of the Product Matrix
Matrix multiplication is performed by taking the dot product of each row of the first matrix with each column of the second matrix. For a product \(C = AB\), the entry in the \(i\)-th row and \(j\)-th column, denoted as \(c_{ij}\), is calculated as:
\[c_{ij} = (\text{row } i \text{ of } A) \cdot (\text{column } j \text{ of } B)\]
We calculate each entry step-by-step:
- For \(c_{11}\): Multiply the first row of \(A\) \([1, -1]\) by the first column of \(B\) \([0, 1]^T\):
\[c_{11} = (1 \times 0) + (-1 \times 1) = 0 - 1 = -1\]
- For \(c_{12}\): Multiply the first row of \(A\) \([1, -1]\) by the second column of \(B\) \([2, -1]^T\):
\[c_{12} = (1 \times 2) + (-1 \times -1) = 2 + 1 = 3\]
- For \(c_{21}\): Multiply the second row of \(A\) \([0, 3]\) by the first column of \(B\) \([0, 1]^T\):
\[c_{21} = (0 \times 0) + (3 \times 1) = 0 + 3 = 3\]
- For \(c_{22}\): Multiply the second row of \(A\) \([0, 3]\) by the second column of \(B\) \([2, -1]^T\):
\[c_{22} = (0 \times 2) + (3 \times -1) = 0 - 3 = -3\]
Thus, the product matrix is \(C = \begin{bmatrix} -1 & 3 \ 3 & -3 \end{bmatrix}\).
Examples
In computer graphics and game development, matrices are used to perform transformations like rotations, scaling, and reflections on 3D models. By multiplying a matrix representing an object's position with a transformation matrix, developers can compute the new coordinates of the object efficiently. Understanding how to compute products of matrices is fundamental to building these systems that allow characters to move and interact within a virtual environment.