- Identify that matrices A and B are both \(2 \times 2\), making element-wise subtraction possible.
- Subtract corresponding elements: \(A - B = \begin{pmatrix} 5 - (-5) & 0 - 5 \ 0 - (-5) & -5 - 5 \end{pmatrix}\).
- Calculate the values for each position: \(10, -5, 5, -10\).
- Final resulting matrix: \(\boxed{\begin{pmatrix} 10 & -5 \ 5 & -10 \end{pmatrix}}\).
Explanation
- Problem Analysis and Definition of Matrix Subtraction
We are given two \(2 \times 2\) matrices:
\[A = \begin{pmatrix} 5 & 0 \ 0 & -5 \end{pmatrix} \quad B = \begin{pmatrix} -5 & 5 \ -5 & 5 \end{pmatrix}\]
Our goal is to compute the difference \(A - B\). Matrix subtraction is defined for matrices of the same dimensions, and it is performed element-wise. That is, if \(A = [a_{ij}]\) and \(B = [b_{ij}]\), then \(A - B = [a_{ij} - b_{ij}]\). Both matrices given are \(2 \times 2\), so subtraction is well-defined.
- Performing Matrix Subtraction
To find the difference \(A - B\), we subtract each corresponding element of matrix \(B\) from matrix \(A\):
\[A - B = \begin{pmatrix} 5 - (-5) & 0 - 5 \ 0 - (-5) & -5 - 5 \end{pmatrix}\]
Now, we perform the arithmetic operations for each entry:
- For the first row, first column: \(5 - (-5) = 5 + 5 = 10\)
- For the first row, second column: \(0 - 5 = -5\)
- For the second row, first column: \(0 - (-5) = 0 + 5 = 5\)
- For the second row, second column: \(-5 - 5 = -10\)
- Final Result Construction
Substituting the results of our calculations into the matrix structure, we obtain:
\[A - B = \begin{pmatrix} 10 & -5 \ 5 & -10 \end{pmatrix}\]
This is our resulting matrix.
Examples
Matrix operations are fundamental in computer graphics and data analysis. For example, in computer graphics, if you have a set of coordinates represented as a matrix and you want to shift them all by a certain amount, you can perform matrix subtraction to adjust those positions. Similarly, in data science, subtracting one data matrix from another helps in calculating the difference between two experimental states, allowing researchers to isolate the effects of a specific change.