- Compute the product of the two matrices: \(X \cdot A = \begin{bmatrix} 1 & -3 \ 4 & 2 \end{bmatrix} \cdot \begin{bmatrix} \frac{1}{7} & \frac{3}{14} \ -\frac{2}{7} & \frac{1}{14} \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).
- Observe that the resulting matrix is the identity matrix, \(I\).
- Conclude that since \(X \cdot A = I\), \(X\) and \(A\) are inverses.
- The product of the matrices \(\boxed{\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}}\) is the identity matrix. Therefore, \(X\) and \(A\) \(\boxed{\text{are}}\) inverses of each other.
Explanation
- Understanding the Problem and Matrix Multiplication
To determine if the matrix \(X = \begin{bmatrix} 1 & -3 \ 4 & 2 \end{bmatrix}\) is the inverse of the matrix \(A = \begin{bmatrix} \frac{1}{7} & \frac{3}{14} \ -\frac{2}{7} & \frac{1}{14} \end{bmatrix}\), we must check if their product is the identity matrix \(I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). The definition of an inverse matrix for a square matrix \(M\) is a matrix \(M^{-1}\) such that \(M \cdot M^{-1} = I\). We will compute the product \(X \cdot A\) row by row and column by column.
- Performing the Matrix Multiplication
We calculate the elements of the product matrix \(P = X \cdot A = \begin{bmatrix} p_{11} & p_{12} \ p_{21} & p_{22} \end{bmatrix}\):
- For \(p_{11}\): \((1 \times \frac{1}{7}) + (-3 \times -\frac{2}{7}) = \frac{1}{7} + \frac{6}{7} = \frac{7}{7} = 1\)
- For \(p_{12}\): \((1 \times \frac{3}{14}) + (-3 \times \frac{1}{14}) = \frac{3}{14} - \frac{3}{14} = 0\)
- For \(p_{21}\): \((4 \times \frac{1}{7}) + (2 \times -\frac{2}{7}) = \frac{4}{7} - \frac{4}{7} = 0\)
- For \(p_{22}\): \((4 \times \frac{3}{14}) + (2 \times \frac{1}{14}) = \frac{12}{14} + \frac{2}{14} = \frac{14}{14} = 1\)
Thus, the product is \(P = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).
- Conclusion
Since the product of the matrices \(X\) and \(A\) equals the identity matrix \(I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\), it follows that \(X\) and \(A\) are indeed inverses of each other. The product of the matrices \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\) is the identity matrix. Therefore, \(X\) and \(A\) are inverses of each other.
Examples
In computer graphics, matrices are often used to represent transformations such as rotation, scaling, or translation. If a matrix \(A\) represents a specific transformation applied to an image, its inverse \(A^{-1}\) is essential to reverse that transformation, effectively returning the image to its original state. By confirming whether a given matrix is an inverse, we ensure that coordinate mapping can be accurately undone in software development, physics engines, and data processing.